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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285741550

Chapter 12.3, Problem 42E

Textbook Problem

Find the scalar and vector projections of **b** onto **a.**

**42. a** = ⟨−1, 4, 8⟩, **b** = ⟨12, 1, 2⟩

Expert Solution

To determine

**To find:** The scalar projection of **b** onto **a** and the vector projection of **b** onto **a**.

**Given:**

**Formula:**

Write the expression for scalar projection of **b** onto **a**.

Consider a general expression to find the dot product between two three-dimensional vectors.

Consider a general expression to find the magnitude of a three dimensional vector that is

Write the expression for vector projection of **b** onto **a**.

In equation (2), substitute –1 for

In equation (3), substitute –1 for

Calculus: Early Transcendentals

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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 |...

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Multiply or Divide Perform the multiplication or division and simplify. 31.x2+7x+12x2+3x+2x2+5x+6x2+6x+9

Precalculus: Mathematics for Calculus (Standalone Book)

One to-One Functions Can the graph of a one-to-one function intersect a horizontal line more than once? Explain...

Calculus: Early Transcendental Functions

SOC/CJ A random sample of 113 felons convicted of nonviolent crimes in a state prison system completed a progra...

Essentials Of Statistics

Find the median for the distribution of scores in the following frequency distribution table. X f 6 1 5 1 4 4 3...

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

1-40 Use the guidelines of this section to sketch the curve. y=1x24

Calculus (MindTap Course List)

Crickets The mountain retreat near Puerto Vallarta was so quiet at night that all one could hear was the chirpi...

Applied Calculus

A normal distribution has a mean of =65 and a standard deviation of =20 . For each of the following samples, co...

Statistics for The Behavioral Sciences (MindTap Course List)

Average Speed of a Vehicle The average speed of a vehicle in miles per hour on a stretch of Route 134 between 6...

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

Use continuity to evaluate the limit. 38. limx2x3/x2+x2

Single Variable Calculus

Let X = the number of nonzero digits in a randomly selected 4-digit PIN that has no restriction on the digits. ...

Probability and Statistics for Engineering and the Sciences

Survey: Manipulation The NewYork Times did a special report on polling that was earned in papers across the nat...

Understanding Basic Statistics

Expand each expression in Exercises 122. (2x3)2

Finite Mathematics

In Exercise 17 to 22, use either the sine ratio or the cosine ratio to find the length of the indicated sides o...

Elementary Geometry For College Students, 7e

For Problems 51-66, use an algebraic approach to solve each problem. Objective 2 The difference of two numbers ...

Intermediate Algebra

In Exercises 19 to 26, show that the given set has a cardinality of K by establishing a one-to-one corresponden...

Mathematical Excursions (MindTap Course List)

1. Find the slope of the line throughand.

Mathematical Applications for the Management, Life, and Social Sciences

Divide (use the remainder form with r): 5308,736

Elementary Technical Mathematics

Related Rates In your own words, state the guidelines for solving related-rate problems.

Calculus of a Single Variable

Suppose that f is continuous on [0, 4], f(0) = 1, and 2 f(x) 5 for all x in (0, 4). Show that 9 f(4) 21.

Single Variable Calculus: Early Transcendentals

If sinA=513 with A in QII, find the following. sinA2

Trigonometry (MindTap Course List)

In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplicat...

Elements Of Modern Algebra

47. Springfield Aggressive fund has total assets of $25,000,000 and liabilities of $1,500,000. If there are 2,6...

Contemporary Mathematics for Business & Consumers

Sometimes, Always, or Never:
If for all n and both {an} and {cn} converge, then{bn} converges.

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

True or False:
The Alternating Series Test may be applied to determine the convergence of

Study Guide for Stewart's Multivariable Calculus, 8th

Finding a Limit In Exercises 65-70, find the limit (if it exists). limt0(t2i+3tj+1costtk)

Calculus: Early Transcendental Functions (MindTap Course List)

In Exercises 1-18, construct a truth table for each compound proposition. (pq)(pr)

Finite Mathematics for the Managerial, Life, and Social Sciences

In Exercises 19 to 24, classify each statement as true or false. Rain is wet and snow is cold.

Elementary Geometry for College Students

Finding Points of Inflection In Exercises 17-32, find the points of inflection and discuss the concavity of the...

Calculus (MindTap Course List)

Cycloidal MotionIn Exercises 45 and 46, consider the motion of a point (or particle) on the circumference of a ...

Multivariable Calculus

A production process is checked periodically by a quality control inspector. The inspector selects simple rando...

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

The percentage of adult users of the Internet who use Facebook has increased over time (Pew Research Internet P...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

A researcher would like to use a factorial study to compare two programs designed to help people stop smoking. ...

Research Methods for the Behavioral Sciences (MindTap Course List)

How is the descriptive strategy different from the other four research strategies?

Research Methods for the Behavioral Sciences (MindTap Course List)

Solve the following exercises. For customary unit-dimensioned exercises, calculate angles to the nearer minute ...

Mathematics For Machine Technology

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If t...

College Algebra (MindTap Course List)

Each person in a large sample of German adolescents was asked to indicate which of 50 popular movies they had s...

Introduction To Statistics And Data Analysis

29. Use the Hill-Huntington Method to apportion the legislative seats in Exercise 5. 5. Use the Hill-Huntington...

Mathematics: A Practical Odyssey

In Problems 120 solve the given system of differential equations by systematic elimination. 17. Dx=yDy=zDz=x

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

A sample of 16 items from population 1 has a sample variance and a sample of 21 items from population 2 has a ...

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

Rewrite the following statements less formally, Without using variables. Determine, as best as you can, whether...

Discrete Mathematics With Applications

36. When a new machine is functioning properly, only 3% of the items produced are defective. Assume that we wil...

Essentials Of Statistics For Business & Economics

Use the quadratic equation to solve 2x2+3x+4=0.

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

For the following exercises, points P( 1.5, 0) and Q( , y) are on the graph of the function f()=cos() . 12. Us...

Calculus Volume 1

In the following exercises, compute each integral using appropriate substitutions. 425. dtt 1 In 2 t

Calculus Volume 2