- 12.1: In Exercises 14, let and Find (a) the component form of the vector ...
- 12.2: In Exercises 14, let and Find (a) the component form of the vector ...
- 12.3: In Exercises 14, let and Find (a) the component form of the vector ...
- 12.4: In Exercises 14, let and Find (a) the component form of the vector ...
- 12.5: In Exercises 58, find the component form of the vector
- 12.6: In Exercises 58, find the component form of the vector
- 12.7: In Exercises 58, find the component form of the vector
- 12.8: In Exercises 58, find the component form of the vector
- 12.9: Express the vectors in Exercises 912 in terms of their lengths and ...
- 12.10: Express the vectors in Exercises 912 in terms of their lengths and ...
- 12.11: Express the vectors in Exercises 912 in terms of their lengths and ...
- 12.12: Express the vectors in Exercises 912 in terms of their lengths and ...
- 12.13: Express the vectors in Exercises 13 and 14 in terms of their length...
- 12.14: Express the vectors in Exercises 13 and 14 in terms of their length...
- 12.15: Find a vector 2 units long in the direction of
- 12.16: Find a vector 5 units long in the direction opposite to the direction
- 12.17: In Exercises 17 and 18, find the angle between v and u, the scalar ...
- 12.18: In Exercises 17 and 18, find the angle between v and u, the scalar ...
- 12.19: In Exercises 19 and 20, find projv u
- 12.20: In Exercises 19 and 20, find projv u
- 12.21: In Exercises 21 and 22, draw coordinate axes and then sketch u, v, ...
- 12.22: In Exercises 21 and 22, draw coordinate axes and then sketch u, v, ...
- 12.23: If and the angle between v and w is find
- 12.24: For what value or values of a will the vectors and be v = -4i - 8j ...
- 12.25: In Exercises 25 and 26, find (a) the area of the parallelogram dete...
- 12.26: In Exercises 25 and 26, find (a) the area of the parallelogram dete...
- 12.27: Suppose that n is normal to a plane and that v is parallel to the p...
- 12.28: Find a vector in the plane parallel to the line
- 12.29: In Exercises 29 and 30, find the distance from the point to the line.
- 12.30: In Exercises 29 and 30, find the distance from the point to the line.
- 12.31: Parametrize the line that passes through the point (1, 2, 3) parall...
- 12.32: Parametrize the line segment joining the points P(1, 2, 0) and
- 12.33: In Exercises 33 and 34, find the distance from the point to the plane.
- 12.34: In Exercises 33 and 34, find the distance from the point to the plane.
- 12.35: Find an equation for the plane that passes through the point normal...
- 12.36: Find an equation for the plane that passes through the point perpen...
- 12.37: In Exercises 37 and 38, find an equation for the plane through poin...
- 12.38: In Exercises 37 and 38, find an equation for the plane through poin...
- 12.39: Find the points in which the line meets the three coordinate planes
- 12.40: Find the point in which the line through the origin perpendicular t...
- 12.41: Find the acute angle between the planes and
- 12.42: Find the acute angle between the planes and
- 12.43: Find parametric equations for the line in which the planes and inte...
- 12.44: Show that the line in which the planes intersect is parallel to the...
- 12.45: The planes and intersect in a line. a. Show that the planes are ort...
- 12.46: Find an equation for the plane that passes through the point (1, 2,...
- 12.47: Is related in any special way to the plane Give reasons for your an...
- 12.48: The equation represents the plane through normal to n. What set doe...
- 12.49: Find the distance from the point P(1, 4, 0) to the plane through A(...
- 12.50: Find the distance from the point (2, 2, 3) to the plane
- 12.51: Find a vector parallel to the plane and orthogonal to
- 12.52: Find a unit vector orthogonal to A in the plane of B and C if
- 12.53: Find a vector of magnitude 2 parallel to the line of intersection o...
- 12.54: Find the point in which the line through the origin perpendicular t...
- 12.55: Find the point in which the line through P(3, 2, 1) normal to the p...
- 12.56: What angle does the line of intersection of the planes and make wit...
- 12.57: The line intersects the plane in a point P. Find the coordinates of...
- 12.58: Show that for every real number k the plane contains the line of in...
- 12.59: Find an equation for the plane through and that lies parallel to th...
- 12.60: Is the line related in any way to the plane Give reasons for your a...
- 12.61: Which of the following are equations for the plane through the poin...
- 12.62: The parallelogram shown here has vertices at and D. Find a. the coo...
- 12.63: Find the distance between the line through the points and and the l...
- 12.64: (Continuation of Exercise 63.) Find the distance between the line t...
- 12.65: Identify and sketch the surfaces in Exercises 6576.
- 12.66: Identify and sketch the surfaces in Exercises 6576.
- 12.67: Identify and sketch the surfaces in Exercises 6576.
- 12.68: Identify and sketch the surfaces in Exercises 6576.
- 12.69: Identify and sketch the surfaces in Exercises 6576.
- 12.70: Identify and sketch the surfaces in Exercises 6576.
- 12.71: Identify and sketch the surfaces in Exercises 6576.
- 12.72: Identify and sketch the surfaces in Exercises 6576.
- 12.73: Identify and sketch the surfaces in Exercises 6576.
- 12.74: Identify and sketch the surfaces in Exercises 6576.
- 12.75: Identify and sketch the surfaces in Exercises 6576.
- 12.76: Identify and sketch the surfaces in Exercises 6576.
Solutions for Chapter 12: Vectors and the Geometry of Space
Full solutions for Thomas' Calculus Early Transcendentals | 12th Edition
ISBN: 9780321588760
Solutions for Chapter 12
20
2
Thomas' Calculus Early Transcendentals was written by and is associated to the ISBN: 9780321588760. This textbook survival guide was created for the textbook: Thomas' Calculus Early Transcendentals, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 12: Vectors and the Geometry of Space includes 76 full step-by-step solutions. Since 76 problems in chapter 12: Vectors and the Geometry of Space have been answered, more than 7322 students have viewed full step-by-step solutions from this chapter.
Key Calculus Terms and definitions covered in this textbook
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Arccotangent function
See Inverse cotangent function.
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Associative properties
a + (b + c) = (a + b) + c, a(bc) = (ab)c.
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Circular functions
Trigonometric functions when applied to real numbers are circular functions
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Complex conjugates
Complex numbers a + bi and a - bi
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Conjugate axis of a hyperbola
The line segment of length 2b that is perpendicular to the focal axis and has the center of the hyperbola as its midpoint
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De Moivre’s theorem
(r(cos ? + i sin ?))n = r n (cos n? + i sin n?)
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Degree
Unit of measurement (represented by the symbol ) for angles or arcs, equal to 1/360 of a complete revolution
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Difference identity
An identity involving a trigonometric function of u - v
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Direction of an arrow
The angle the arrow makes with the positive x-axis
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Distance (in a coordinate plane)
The distance d(P, Q) between P(x, y) and Q(x, y) d(P, Q) = 2(x 1 - x 2)2 + (y1 - y2)2
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Domain of validity of an identity
The set of values of the variable for which both sides of the identity are defined
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Gaussian elimination
A method of solving a system of n linear equations in n unknowns.
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Polar distance formula
The distance between the points with polar coordinates (r1, ?1 ) and (r2, ?2 ) = 2r 12 + r 22 - 2r1r2 cos 1?1 - ?22
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Quadratic equation in x
An equation that can be written in the form ax 2 + bx + c = 01a ? 02
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Root of a number
See Principal nth root.
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Semiperimeter of a triangle
One-half of the sum of the lengths of the sides of a triangle.
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Tangent
The function y = tan x
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Transformation
A function that maps real numbers to real numbers.
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Vector equation for a line in space
The line through P0(x 0, y0, z0) in the direction of the nonzero vector V = <a, b, c> has vector equation r = r0 + tv , where r = <x,y,z>.
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Zero factorial
See n factorial.