Math
CalculusMultivariable CalculusFind two unit vectors that are orthogonal to both j + 2 k and i − 2 j + 3 k .

Find two unit vectors that are orthogonal to both j + 2 k and i − 2 j + 3 k .

BuyFind

Multivariable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266643
BuyFind

Multivariable Calculus

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781305266643

Solutions

Chapter
Section
Problem 1RCC
Problem 2RCC
Problem 3RCC
Problem 4RCC
Problem 5RCC
Problem 6RCC
Problem 7RCC
Problem 8RCC
Problem 9RCC
Problem 10RCC
Problem 11RCC
Problem 12RCC
Problem 13RCC
Problem 14RCC
Problem 15RCC
Problem 16RCC
Problem 17RCC
Problem 18RCC
Problem 19RCC
Problem 1RQ
Problem 2RQ
Problem 3RQ
Problem 4RQ
Problem 5RQ
Problem 6RQ
Problem 7RQ
Problem 8RQ
Problem 9RQ
Problem 10RQ
Problem 11RQ
Problem 12RQ
Problem 13RQ
Problem 14RQ
Problem 15RQ
Problem 16RQ
Problem 17RQ
Problem 18RQ
Problem 19RQ
Problem 20RQ
Problem 21RQ
Problem 22RQ
Problem 1RE
Problem 2RE
Problem 3RE
Problem 4RE
Problem 5RE
Problem 6RE
Problem 7RE
Problem 8RE
Problem 9RE
Problem 10RE
Problem 11RE
Problem 12RE
Problem 13RE
Problem 14RE
Problem 15RE
Problem 16RE
Problem 17RE
Problem 18RE
Problem 19RE
Problem 20RE
Problem 21RE
Problem 22RE
Problem 23RE
Problem 24RE
Problem 25RE
Problem 26RE
Problem 27RE
Problem 28RE
Problem 29RE
Problem 30RE
Problem 31RE
Problem 32RE
Problem 33RE
Problem 34RE
Problem 35RE
Problem 36RE
Problem 37RE
Problem 38RE
Problem 1P
Problem 2P
Problem 5P
Chapter 12, Problem 6RE
Textbook Problem

Find two unit vectors that are orthogonal to both j + 2k and i − 2j + 3k.

Expert Solution

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Chapter 12,
Problem 5RE
Chapter 12,
Problem 7RE

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Chapter 12 Solutions

Multivariable Calculus
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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