Math

CalculusMultivariable CalculusSuppose the three coordinate planes are all mirrored and a light ray given by the vector a = ⟨ a 1 , a 2 , a 3 ⟩ first strikes the xz -plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by b = ⟨ a 1 , − a 2 , a 3 ⟩. Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the moon, to calculate precisely the distance from the earth to the moon.)BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266643

Chapter 12.2, Problem 52E

Textbook Problem

Suppose the three coordinate planes are all mirrored and a light ray given by the vector **a** = ⟨*a*_{1}, *a*_{2}, *a*_{3}⟩ first strikes the *xz*-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by **b** = ⟨*a*_{1}, −*a*_{2}, *a*_{3}⟩. Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the moon, to calculate precisely the distance from the earth to the moon.)

Multivariable Calculus

Show all chapter solutions

Ch. 12.1 - Suppose you start at the origin, move along the...Ch. 12.1 - Sketch the points (1, 5, 3), (0, 2, 3), (3, 0, 2),...Ch. 12.1 - Which of the points A(4, 0, 1), B(3, 1, 5), and...Ch. 12.1 - What are the projections of the point (2. 3. 5) on...Ch. 12.1 - What does the equation x = 4 represent in 2? What...Ch. 12.1 - What does the equation y = 3 represent in 3? What...Ch. 12.1 - Describe and sketch the surface in 3 represented...Ch. 12.1 - Describe and sketch the surface in 3 represented...Ch. 12.1 - Find the lengths of the sides of the triangle PQR....Ch. 12.1 - Find the lengths of the sides of the triangle PQR....

Ch. 12.1 - Determine whether the points lie on a straight...Ch. 12.1 - Find the distance from (4, 2, 6) to each of the...Ch. 12.1 - Find an equation of the sphere with center (3, 2,...Ch. 12.1 - Find an equation of the sphere with center (2, 6,...Ch. 12.1 - Find an equation of the sphere that passes through...Ch. 12.1 - Find an equation of the sphere that passes through...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - Show that the equation represents a sphere, and...Ch. 12.1 - (a) Prove that the midpoint of the line segment...Ch. 12.1 - Find an equation of a sphere if one of its...Ch. 12.1 - Find equations of the spheres with center (2, 3,...Ch. 12.1 - Find an equation of the largest sphere with center...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Describe in words the region of 3 represented by...Ch. 12.1 - Write inequalities to describe the region. 39. The...Ch. 12.1 - Write inequalities to describe the region. 40. The...Ch. 12.1 - Write inequalities to describe the region. 41. The...Ch. 12.1 - Write inequalities to describe the region. 42. The...Ch. 12.1 - The figure shows a line L1 in space and a second...Ch. 12.1 - Consider the points P such that the distance from...Ch. 12.1 - Find an equation of the set of all points...Ch. 12.1 - Find the volume of the solid that lies inside both...Ch. 12.1 - Find the distance between the spheres x2 + y2 + z2...Ch. 12.1 - Describe and sketch a solid with the following...Ch. 12.2 - Are the following quantities vectors or scalars?...Ch. 12.2 - What is the relationship between the point (4, 7)...Ch. 12.2 - Name all the equal vectors in the parallelogram...Ch. 12.2 - Write each combination of vectors as a single...Ch. 12.2 - Copy the vectors in the figure and use them to...Ch. 12.2 - Copy the vectors in the figure and use them to...Ch. 12.2 - In the figure, the tip of c and the tail of d are...Ch. 12.2 - If the vectors in the figure satisfy |u | = | v |...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find a vector a with representation given by the...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find the sum of the given vectors and illustrate...Ch. 12.2 - Find a + b, 4a + 2b, | a |, and | a b | 19. a =...Ch. 12.2 - Find a + b, 4a + 2b, | a |, and | a b | 20. a =...Ch. 12.2 - Find a + b, 4a + 2b, | a |, and | a b | 21. a =...Ch. 12.2 - Find a + b, 4a + 2b, | a |, and | a b | 22. a =...Ch. 12.2 - Find a unit vector that has the same direction as...Ch. 12.2 - Find a unit vector that has the same direction as...Ch. 12.2 - Find a unit vector that has the same direction as...Ch. 12.2 - Find the vector that has the same direction as 6,...Ch. 12.2 - What is the angle between the given vector and the...Ch. 12.2 - What is the angle between the given vector and the...Ch. 12.2 - If v lies in the first quadrant and makes an angle...Ch. 12.2 - If a child pulls a sled through the snow on a...Ch. 12.2 - A quarterback throws a football with angle of...Ch. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - Find the magnitude of the resultant force and the...Ch. 12.2 - The magnitude of a velocity vector is called...Ch. 12.2 - A woman walks due west on the deck of a ship at 3...Ch. 12.2 - A crane suspends a 500-lb steel beam horizontally...Ch. 12.2 - A block-and-tackle pulley hoist is suspended in a...Ch. 12.2 - The tension T at each end of a chain has magnitude...Ch. 12.2 - A boatman wants to cross a canal that is 3 km wide...Ch. 12.2 - Three forces act on an object. Two of the forces...Ch. 12.2 - Find the unit vectors that are parallel to the...Ch. 12.2 - (a) Find the unit vectors that are parallel to the...Ch. 12.2 - If A, B, and C are the vertices of a triangle,...Ch. 12.2 - Let C be the point on the line segment AB that is...Ch. 12.2 - (a) Draw the vectors a = 3, 2, b = 2, 1, and c =...Ch. 12.2 - Suppose that a and b are nonzero vectors that are...Ch. 12.2 - If r = x, y, z and r0 = x0, y0, z0, describe the...Ch. 12.2 - If r = x, y, r1 = x1, y1, and r2 = x2, y2,...Ch. 12.2 - Figure 16 gives a geometric demonstration of...Ch. 12.2 - Prove Property 5 of vectors algebraically for the...Ch. 12.2 - Use vectors to prove that the line joining the...Ch. 12.2 - Suppose the three coordinate planes are all...Ch. 12.3 - Which of the following expressions are meaningful?...Ch. 12.3 - Find a b. 2. a = 5, 2, b = 3, 4Ch. 12.3 - Find a b. 3. a = 1.5, 0.4, b = 4, 6Ch. 12.3 - Find a b. 4. a = 6, 2, 3, b = 2, 5, 1Ch. 12.3 - Find a b. 5. a = 4, 1, 14, b = 6, 3, 8Ch. 12.3 - Find a b. 6. a. = p, p, 2p, b = 2q, q, qCh. 12.3 - Find a b. 7. a = 2i + j, b = i j + kCh. 12.3 - Find a b. 8. a = 3i + 2j k, b = 4i + 5kCh. 12.3 - Find a b. 9. | a | = 7, | b | = 4, the angle...Ch. 12.3 - Find a b. 10. | a | = 80, | b | = 50, the angle...Ch. 12.3 - If u is a unit vector, find u v and u w.Ch. 12.3 - If u is a unit vector, find u v and u w.Ch. 12.3 - (a) Show that i j = j k = k i = 0. (b) Show...Ch. 12.3 - A street vendor sells a hamburgers, b hot dogs,...Ch. 12.3 - Find the angle between the vectors. (First find an...Ch. 12.3 - Find the angle between the vectors. (First find an...Ch. 12.3 - Find the angle between the vectors. (First find an...Ch. 12.3 - Find the angle between the vectors. (First find an...Ch. 12.3 - Find the angle between the vectors. (First find an...Ch. 12.3 - Find the angle between the vectors. (First find an...Ch. 12.3 - Find, correct to the nearest degree, the three...Ch. 12.3 - Find, correct to the nearest degree, the three...Ch. 12.3 - Determine whether the given vectors are...Ch. 12.3 - Determine whether the given vectors are...Ch. 12.3 - Use vectors to decide whether the triangle with...Ch. 12.3 - Find the values of x such that the angle between...Ch. 12.3 - Find a unit vector that is orthogonal to both i +...Ch. 12.3 - Find two unit vectors that make an angle of 60...Ch. 12.3 - Find the acute angle between the lines. 29. 2x y...Ch. 12.3 - Find the acute angle between the lines. 30. x + 2y...Ch. 12.3 - Find the acute angles between the curves at their...Ch. 12.3 - Find the acute angles between the curves at their...Ch. 12.3 - Find the direction cosines and direction angles of...Ch. 12.3 - Find the direction cosines and direction angles of...Ch. 12.3 - Find the direction cosines and direction angles of...Ch. 12.3 - Find the direction cosines and direction angles of...Ch. 12.3 - Find the direction cosines and direction angles of...Ch. 12.3 - If a vector has direction angles = /4 and = /3,...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Find the scalar and vector projections of b onto...Ch. 12.3 - Show that the vector ortha b = b proja b is...Ch. 12.3 - For the vectors in Exercise 40, find ortha b and...Ch. 12.3 - If a = 3, 0, 1, find a vector b such that compa b...Ch. 12.3 - Suppose that a and b are nonzero vectors. (a)...Ch. 12.3 - Find the work done by a force F = 8i 6j + 9k that...Ch. 12.3 - A tow truck drags a stalled car along a road. The...Ch. 12.3 - A sled is pulled along a level path through snow...Ch. 12.3 - A boat sails south with the help of a wind blowing...Ch. 12.3 - Use a scalar projection to show that the distance...Ch. 12.3 - If r = x, y, z a = a1, a2, a3, and b = b1, b2, b3,...Ch. 12.3 - Find the angle between a diagonal of a cube and...Ch. 12.3 - Find the angle between a diagonal of a cube and a...Ch. 12.3 - A molecule of methane, CH4, is structured with the...Ch. 12.3 - If c = | a | b + | b | a, where a, b, and c are...Ch. 12.3 - Prove Properties 2.4, and 5 of the dot product...Ch. 12.3 - Suppose that all sides of a quadrilateral are...Ch. 12.3 - Use Theorem 3 to prove the Cauchy-Schwarz...Ch. 12.3 - The Triangle Inequality for vectors is | a + b | ...Ch. 12.3 - The Parallelogram Law states that | a + b |2 + | a...Ch. 12.3 - Show that if u + v and u v are orthogonal, then...Ch. 12.3 - If is the angle between vectors a and b. show...Ch. 12.4 - Find the cross product a b and verify that it is...Ch. 12.4 - Find the cross product a b and verify that it is...Ch. 12.4 - Find the cross product a b and verify that it is...Ch. 12.4 - Find the cross product a b and verify that it is...Ch. 12.4 - Find the cross product a b and verify that it is...Ch. 12.4 - Find the cross product a b and verify that it is...Ch. 12.4 - Find the cross product a b and verify that it is...Ch. 12.4 - If a = i 2k and b = j + k, find a b. Sketch a,...Ch. 12.4 - Find the vector, not with determinants, but by...Ch. 12.4 - Find the vector, not with determinants, but by...Ch. 12.4 - Find the vector, not with determinants, but by...Ch. 12.4 - Find the vector, not with determinants, but by...Ch. 12.4 - State whether each expression is meaningful. If...Ch. 12.4 - Find |u v| and determine whether u v is directed...Ch. 12.4 - Find |u v| and determine whether u v is directed...Ch. 12.4 - The figure shows a vector a in the xy-plane and a...Ch. 12.4 - If a = 2, 1, 3 and b = 4, 2, 1, find a b and b ...Ch. 12.4 - If a = 1, 0, 1, b = 2, 1, 1. and c = 0, 1, 3, show...Ch. 12.4 - Find two unit vectors orthogonal to both 3, 2, 1...Ch. 12.4 - Find two unit vectors orthogonal to both j k and...Ch. 12.4 - Show that 0 a = 0 = a 0 for any vector a in V3.Ch. 12.4 - Show that (a b) b = 0 for all vectors a and b in...Ch. 12.4 - Prove the property of cross products (Theorem 11)....Ch. 12.4 - Prove the property of cross products (Theorem 11)....Ch. 12.4 - Prove the property of cross products (Theorem 11)....Ch. 12.4 - Prove the property of cross products (Theorem 11)....Ch. 12.4 - Find the area of the parallelogram with vertices...Ch. 12.4 - Find the area of the parallelogram with vertices...Ch. 12.4 - (a) Find a nonzero vector orthogonal to the plane...Ch. 12.4 - (a) Find a nonzero vector orthogonal to the plane...Ch. 12.4 - (a) Find a nonzero vector orthogonal to the plane...Ch. 12.4 - (a) Find a nonzero vector orthogonal to the plane...Ch. 12.4 - Find the volume of the parallelepiped determined...Ch. 12.4 - Find the volume of the parallelepiped determined...Ch. 12.4 - Find the volume of the parallelepiped with...Ch. 12.4 - Find the volume of the parallelepiped with...Ch. 12.4 - Use the scalar triple product to verify that the...Ch. 12.4 - Use the scalar triple product to determine whether...Ch. 12.4 - A bicycle pedal is pushed by a foot with a 60-N...Ch. 12.4 - (a) A horizontal force of 20 lb is applied to the...Ch. 12.4 - A wrench 30 cm long lies along the positive y-axis...Ch. 12.4 - Let v = 5j and let u be a vector with length 3...Ch. 12.4 - If a b = 3 and a b = 1, 2, 2, find the angle...Ch. 12.4 - (a) Find all vectors v such that 1, 2, 1 x v = 3,...Ch. 12.4 - (a) Let P be a point not on the line L that passes...Ch. 12.4 - Show that | a b |2 = | a |2| b |2 (a b)2Ch. 12.4 - If a + b + c = 0. show that a b = b c = c aCh. 12.4 - Prove that (a b) (a + b) = 2(a b).Ch. 12.4 - Prove Properly 6 of cross products, that is, a (b...Ch. 12.4 - Use Exercise 50 to prove that a (b c) + b (c ...Ch. 12.4 - Prove that (ab)(cd)=|acbcadbd|Ch. 12.4 - Suppose that a 0. (a) If a b = a c, does it...Ch. 12.4 - If v1, v2, and v3 are noncoplanar vectors, let...Ch. 12.5 - Determine whether each statement is true or false...Ch. 12.5 - Find a vector equation and parametric equations...Ch. 12.5 - Find a vector equation and parametric equations...Ch. 12.5 - Find a vector equation and parametric equations...Ch. 12.5 - Find a vector equation and parametric equations...Ch. 12.5 - Find parametric equations and symmetric equations...Ch. 12.5 - Find parametric equations and symmetric equations...Ch. 12.5 - Find parametric equations and symmetric equations...Ch. 12.5 - Find parametric equations and symmetric equations...Ch. 12.5 - Find parametric equations and symmetric equations...Ch. 12.5 - Find parametric equations and symmetric equations...Ch. 12.5 - Find parametric equations and symmetric equations...Ch. 12.5 - Is the line through (4, 6, 1) and (2, 0, 3)...Ch. 12.5 - Is the line through (2, 4, 0) and (1, 1, 1)...Ch. 12.5 - (a) Find symmetric equations for the line that...Ch. 12.5 - (a) Find parametric equations for the line through...Ch. 12.5 - Find a vector equation for the line segment from...Ch. 12.5 - Find parametric equations for the line segment...Ch. 12.5 - Determine whether the lines L1 and L2 are...Ch. 12.5 - Determine whether the lines L1 and L2 are...Ch. 12.5 - Determine whether the lines L1 and L2 are...Ch. 12.5 - Determine whether the lines L1 and L2 are...Ch. 12.5 - Find an equation of the plane. 23. The plane...Ch. 12.5 - Find an equation of the plane. 24. The plane...Ch. 12.5 - Find an equation of the plane. 25. The plane...Ch. 12.5 - Find an equation of the plane. 26. The plane...Ch. 12.5 - Find an equation of the plane. 27. The plane...Ch. 12.5 - Find an equation of the plane. 28. The plane...Ch. 12.5 - Find an equation of the plane. 29. The plane...Ch. 12.5 - Find an equation of the plane. 30. The plane that...Ch. 12.5 - Find an equation of the plane. 31. The plane...Ch. 12.5 - Find an equation of the plane. 32. The plane...Ch. 12.5 - Find an equation of the plane. 33. The plane...Ch. 12.5 - Find an equation of the plane. 34. The plane...Ch. 12.5 - Find an equation of the plane. 35. The plane that...Ch. 12.5 - Find an equation of the plane. 36. The plane that...Ch. 12.5 - Find an equation of the plane. 37. The plane that...Ch. 12.5 - Find an equation of the plane. 38. The plane that...Ch. 12.5 - Find an equation of the plane. 39. The plane that...Ch. 12.5 - Find an equation of the plane. 40. The plane that...Ch. 12.5 - Use intercepts to help sketch the plane. 41. 2x +...Ch. 12.5 - Use intercepts to help sketch the plane. 42. 3x +...Ch. 12.5 - Use intercepts to help sketch the plane. 43. 6x ...Ch. 12.5 - Use intercepts to help sketch the plane. 44. 6x +...Ch. 12.5 - Find the point at which the line intersects the...Ch. 12.5 - Find the point at which the line intersects the...Ch. 12.5 - Find the point at which the line intersects the...Ch. 12.5 - Where does the line through (3, 1, 0) and (1, 5,...Ch. 12.5 - Find direction numbers for the line of...Ch. 12.5 - Find the cosine of the angle between the planes x...Ch. 12.5 - Determine whether the planes are parallel,...Ch. 12.5 - Determine whether the planes are parallel,...Ch. 12.5 - Determine whether the planes are parallel,...Ch. 12.5 - Determine whether the planes are parallel,...Ch. 12.5 - Determine whether the planes are parallel,...Ch. 12.5 - Determine whether the planes are parallel,...Ch. 12.5 - (a) Find parametric equations for the line of...Ch. 12.5 - (a) Find parametric equations for the line of...Ch. 12.5 - Find symmetric equations for the line of...Ch. 12.5 - Find symmetric equations for the line of...Ch. 12.5 - Find an equation for the plane consisting of all...Ch. 12.5 - Find an equation for the plane consisting of all...Ch. 12.5 - Find an equation of the plane with x-intercept a,...Ch. 12.5 - (a) Find the point at which the given lines...Ch. 12.5 - Find parametric equations for the line through the...Ch. 12.5 - Find parametric equations for the line through the...Ch. 12.5 - Which of the following four planes are parallel?...Ch. 12.5 - Which of the following four lines are parallel?...Ch. 12.5 - Use the formula in Exercise 12.4.45 to find the...Ch. 12.5 - Use the formula in Exercise 12.4.45 to find the...Ch. 12.5 - Find the distance from the point to the given...Ch. 12.5 - Find the distance from the point to the given...Ch. 12.5 - Find the distance between the given parallel...Ch. 12.5 - Find the distance between the given parallel...Ch. 12.5 - Show that the distance between the parallel planes...Ch. 12.5 - Find equations of the planes that are parallel to...Ch. 12.5 - Show that the lines with symmetric equations x = y...Ch. 12.5 - Find the distance between the skew lines with...Ch. 12.5 - Let L1 be the line through the origin and the...Ch. 12.5 - Let L1 be the line through the points (1, 2, 6)...Ch. 12.5 - Two tanks are participating in a battle...Ch. 12.5 - Give a geometric description of each family of...Ch. 12.5 - If a, b, and c are not all 0, show that the...Ch. 12.6 - (a) What does the equation y = x2 represent as a...Ch. 12.6 - (a)Sketch the graph of y = ex as a curve in 2. (b)...Ch. 12.6 - Describe and sketch the surface. x2 + z2 = 1Ch. 12.6 - Describe and sketch the surface. 4x2 + y2 = 4Ch. 12.6 - Describe and sketch the surface. z = 1 y2Ch. 12.6 - Describe and sketch the surface. y = z2Ch. 12.6 - Describe and sketch the surface. xy = 1Ch. 12.6 - Describe and sketch the surface. z = sin yCh. 12.6 - (a) Find and identify the traces of the quadric...Ch. 12.6 - (a) Find and identify the traces of the quadric...Ch. 12.6 - Use traces to sketch and identify the surface. x =...Ch. 12.6 - Use traces to sketch and identify the surface. 4x2...Ch. 12.6 - Use traces to sketch and identify the surface. x2...Ch. 12.6 - Use traces to sketch and identify the surface. z2 ...Ch. 12.6 - Use traces to sketch and identify the surface. 9y2...Ch. 12.6 - Use traces to sketch and identify the surface. 3x2...Ch. 12.6 - Use traces to sketch and identify the surface....Ch. 12.6 - Use traces to sketch and identify the surface. 3x2...Ch. 12.6 - Use traces to sketch and identify the surface. y =...Ch. 12.6 - Use traces to sketch and identify the surface. x =...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Match the equation with its graph (labeled...Ch. 12.6 - Sketch and identify a quadric surface that could...Ch. 12.6 - Sketch and identify a quadric surface that could...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Reduce the equation to one of the standard forms,...Ch. 12.6 - Sketch the region bounded by the surfaces z=x2+y2...Ch. 12.6 - Sketch the region bounded by the paraboloids z =...Ch. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Find an equation for the surface obtained by...Ch. 12.6 - Find an equation for the surface consisting of all...Ch. 12.6 - Find an equation for the surface consisting of all...Ch. 12.6 - Traditionally, the earths surface has been modeled...Ch. 12.6 - A cooling tower for a nuclear reactor is to be...Ch. 12.6 - Show that if the point (a, b, c) lies on the...Ch. 12.6 - Show that the curve of intersection of the...Ch. 12 - What is the difference between a vector and a...Ch. 12 - How do you add two vectors geometrically? How do...Ch. 12 - If a is a vector and c is a scalar, how is ca...Ch. 12 - How do you find the vector from one point to...Ch. 12 - How do you find the dot product a b of two...Ch. 12 - How are dot products useful?Ch. 12 - Write expressions for the scalar and vector...Ch. 12 - How do you find the cross product a b of two...Ch. 12 - How are cross products useful?Ch. 12 - (a) How do you find the area of the parallelogram...Ch. 12 - How do you find a vector perpendicular to a plane?Ch. 12 - How do you find the angle between two intersecting...Ch. 12 - Write a vector equation, parametric equations, and...Ch. 12 - Write a vector equation and a scalar equation for...Ch. 12 - (a) How do you tell if two vectors are parallel?...Ch. 12 - (a) Describe a method for determining whether...Ch. 12 - (a) How do you find the distance from a point to a...Ch. 12 - What are the traces of a surface? How do you find...Ch. 12 - Write equations in standard form of the six types...Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - (a) Find an equation of the sphere that passes...Ch. 12 - Copy the vectors in the figure and use them to...Ch. 12 - If u and v are the vectors shown in the figure,...Ch. 12 - Calculate the given quantity if a = i + j 2k b =...Ch. 12 - Find the values of x such that the vectors 3, 2, x...Ch. 12 - Find two unit vectors that are orthogonal to both...Ch. 12 - Suppose that u (v w) = 2. Find (a) (u v) w (b)...Ch. 12 - Show that if a, b, and c are in V3, then (a b) ...Ch. 12 - Find the acute angle between two diagonals of a...Ch. 12 - Given the points A( 1, 0, 1), B(2, 3, 0), C(l,...Ch. 12 - (a) Find a vector perpendicular to the plane...Ch. 12 - A constant force F = 3i + 5j + 10k moves an object...Ch. 12 - A boat is pulled onto shore using two ropes, as...Ch. 12 - Find the magnitude of the torque about P if a 50-N...Ch. 12 - Find parametric equations for the line. 15. The...Ch. 12 - Find parametric equations for the line. 16. The...Ch. 12 - Find parametric equations for the line. 17. The...Ch. 12 - Find an equation of the plane. 18. The plane...Ch. 12 - Find an equation of the plane. 19. The plane...Ch. 12 - Find an equation of the plane. 20. The plane...Ch. 12 - Find the point in which the line with parametric...Ch. 12 - Find the distance from the origin to the line x =...Ch. 12 - Determine whether the lines given by the symmetric...Ch. 12 - (a) Show that the planes x + y z = 1 and 2x 3y +...Ch. 12 - Find an equation of the plane through the line of...Ch. 12 - (a) Find an equation of the plane that passes...Ch. 12 - Find the distance between the planes 3x + y 4z =...Ch. 12 - Identify and sketch the graph of each surface. 28....Ch. 12 - Identify and sketch the graph of each surface. 29....Ch. 12 - Identify and sketch the graph of each surface. 30....Ch. 12 - Identify and sketch the graph of each surface. 31....Ch. 12 - Identify and sketch the graph of each surface. 32....Ch. 12 - Identify and sketch the graph of each surface. 33....Ch. 12 - Identify and sketch the graph of each surface. 34....Ch. 12 - Identify and sketch the graph of each surface. 35....Ch. 12 - Identify and sketch the graph of each surface. 36....Ch. 12 - An ellipsoid is created by rotating the ellipse...Ch. 12 - A surface consists of all points P such that the...Ch. 12 - Each edge of a cubical box has length 1 m. The box...Ch. 12 - Let B be a solid box with length L. width W, and...Ch. 12 - Suppose v1 and v2 are vectors with |v1| = 2, |v2 |...

Find more solutions based on key concepts

Show solutions True or False? In Exercises 83 and 84, determine whether the statement is true or false. If it is false, explai...

Calculus: An Applied Approach (MindTap Course List)

Tangent Line Let P(5,12) be a point on the circle x2+y2=169 (see figure). (a) What is the slope of the line joi...

Calculus: Early Transcendental Functions

In Exercises 9-14, decide whether the specified values come from a linear, quadratic, exponential, or absolute ...

Applied Calculus

Rational Exponents Simplify the expression and eliminate any negative exponent(s). Assume that all letters deno...

Precalculus: Mathematics for Calculus (Standalone Book)

Find the limit. limx1sin(x1)x2+x2

Single Variable Calculus: Early Transcendentals, Volume I

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or gi...

Calculus: Early Transcendentals

Evaluate the expression sin Exercises 116. (23)2

Finite Mathematics and Applied Calculus (MindTap Course List)

Identify all variables in the research situations listed below and classify them by their level of measurement....

Essentials Of Statistics

In a sample of n 6 scores, five of the scores are each above the mean by one point. Where is the sixth score l...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

2128 Match the equation with reasons for your choice. its graph labeled I-VIII. Give reasons for your choice. y...

Calculus (MindTap Course List)

Law: Corporation Lawsuits Many people say the civil justice system is overburdened. Many cases center on suits ...

Understanding Basic Statistics

What are the two requirements for a random sample?

Statistics for The Behavioral Sciences (MindTap Course List)

Infant Mortality Rates in Massachusetts The deaths of children younger than 1 year old per 1000 live births is ...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Use the table to evaluate each expression. (a) f(g(1)) (b) g(f(1)) (c) f(f(1)) (d) g(g(1)) (e) (g f)(3) (f) (f...

Single Variable Calculus

For Problems 51-78, simplify each of the numerical expressions. Objective 3 (168)4(31)

Intermediate Algebra

The special case of the gamma distribution in which is a positive integer n is called an Erlang distribution. ...

Probability and Statistics for Engineering and the Sciences

In Exercises 1-4, use the graph of the function f to find approximations of the given values. a. f(1) b. f(0) c...

Finite Mathematics

Perform the computations in exercises 112 and express the results in standard form a+bi. 43i4+3i

Elements Of Modern Algebra

In Exercises 51 and 52, draw a Venn diagram with each of the given elements placed in the correct region. U=,,,...

Mathematical Excursions (MindTap Course List)

In problems 1-4, use or to indicate whether the given object is an element of the given set.

Mathematical Applications for the Management, Life, and Social Sciences

Name three types of reasoning.

Elementary Geometry For College Students, 7e

Calculate the interest, purchase price, and effective interest rate of the following Treasury bill (T-bill) pur...

Contemporary Mathematics for Business & Consumers

The sum of two resistors is 550 . One is 4.5 times the other. Find the size of each resistor.

Elementary Technical Mathematics

Find if is between 0 and 90. Round your answers to the nearest tenth of a degree. cot=7.0234

Trigonometry (MindTap Course List)

Finding Volume In Exercises 29-34, set up and evaluate a double integral to find the volume of Lie solid bounde...

Calculus: Early Transcendental Functions (MindTap Course List)

Determine all real numbers a0 for which there exists a nonnegative alive continuous function f(x) defined on 0,...

Calculus of a Single Variable

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. (a) Expr...

Single Variable Calculus: Early Transcendentals

In Exercises 7-12, give the range of values that the random variable X may assume and classify the random varia...

Finite Mathematics for the Managerial, Life, and Social Sciences

It does not exist.

Study Guide for Stewart's Multivariable Calculus, 8th

The quadratic approximation for at a = 1 is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Which block letters have a line symmetry at least one axis? b point symmetry? B H J S W

Elementary Geometry for College Students

The article You Will Be Tested on This (The Chronicle of Higher Education, June 8, 2007) describes an experimen...

Introduction To Statistics And Data Analysis

Radians to Degrees In Exercises 61-64. convert the radian measure to degree measure. 114

Calculus (MindTap Course List)

Finding Volume Using Technology In Exercises 41-44. use a computer algebra system to find the volume of the sol...

Multivariable Calculus

Simplify each expression. 5[62+(95)]4(23)2

College Algebra (MindTap Course List)

Consider the following hypothesis test. H0:12=0Ha:120 The following results are for two independent samples tak...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

A random telephone survey of 1021 adults (aged 18 and older) was conducted by Opinion Research Corporation on b...

Statistics for Business & Economics, Revised (MindTap Course List)

A researcher would like to compare two methods for teaching math to third-grade students. Two third-grade class...

Research Methods for the Behavioral Sciences (MindTap Course List)

Define, compare, and contrast the experimental, nonexperimental, and quasi-experimental research strategies, an...

Research Methods for the Behavioral Sciences (MindTap Course List)

Express the following decimal fractions as common fractions. Reduce to lowest terms. 50. 0.45

Mathematics For Machine Technology

Reminder Round all answers to two decimal places unless otherwise indicated. Radioactive Decay The amount remai...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

In 25—27, find a regular expression that defines the given language. 25. The language consisting of strings of ...

Discrete Mathematics With Applications

Given are five observations collected in a regression study on two variables.
Develop a scatter diagram for th...

Essentials Of Statistics For Business & Economics

Use the Maclaurin series for sin x and cos x along with long division to find the first three nonzero terms of ...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

Al-Noor Koorji bought a house for 189,000. He put 20 down and financed the rest with a thirty-year loan at 534....

Mathematics: A Practical Odyssey

Consider the Costello Music Company problem in exercise 48. The quarterly sales data follow.

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Why is u-substitution referred to as change of variable?

Calculus Volume 2

Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At ...

Introductory Statistics