 12.1: In Exercises 16, let v = 2, 5 and w = 3, 2. Calculate 5w 3v and 5v 3w.
 12.2: In Exercises 16, let v = 2, 5 and w = 3, 2. Sketch v, w, and 2v 3w.
 12.3: In Exercises 16, let v = 2, 5 and w = 3, 2. Find the unit vector in...
 12.4: In Exercises 16, let v = 2, 5 and w = 3, 2. Find the length of v + w.
 12.5: In Exercises 16, let v = 2, 5 and w = 3, 2. Express i as a linear c...
 12.6: In Exercises 16, let v = 2, 5 and w = 3, 2. Find a scalar such that...
 12.7: If P = (1, 4) and Q = (3, 5), what are the components of P Q? What ...
 12.8: Let A = (2, 1), B = (1, 4), and P = (2, 3). Find the point Q such t...
 12.9: Find the vector with length 3 making an angle of 7 4 with the posit...
 12.10: Calculate 3 (i 2j) 6 (i + 6j).
 12.11: Find the value of for which w = 2, is parallel to v = 4, 3.
 12.12: Let P = (1, 4, 3). (a) Find the point Q such that P Q is equivalent...
 12.13: Let w = 2, 2, 1 and v = 4, 5, 4. Solve for u if v + 5u = 3w u.
 12.14: Let v = 3i j + 4k. Find the length of v and the vector 2v + 3 (4i k).
 12.15: Find a parametrization r1(t) of the line passing through (1, 4, 5) ...
 12.16: Let r1(t) = v1 + tw1 and r2(t) = v2 + tw2 be parametrizations of li...
 12.17: Find a and b such that the lines r1 = 1, 2, 1 + t1, 1, 1 and r2 = 3...
 12.18: Find a such that the lines r1 = 1, 2, 1 + t1, 1, 1 and r2 = 3, 1, 1...
 12.19: Sketch the vector sum v = v1 v2 + v3 for the vectors in Figure 1(A).
 12.20: Sketch the sums v1 + v2 + v3, v1 + 2v2, and v2 v3 for the vectors i...
 12.21: In Exercises 2126, let v = 1, 3, 2 and w = 2, 1, 4. 21. Compute v w.
 12.22: Compute the angle between v and w.
 12.23: Compute v w.
 12.24: Find the area of the parallelogram spanned by v and w
 12.25: Find the volume of the parallelepiped spanned by v, w, and u = 1, 2...
 12.26: Find all the vectors orthogonal to both v and w
 12.27: Use vectors to prove that the line connecting the midpoints of two ...
 12.28: Let v = 1, 1, 3 and w = 4, 2, 1. (a) Find the decomposition v = vw ...
 12.29: Calculate the component of v = 2, 1 2 , 3 along w = 1, 2, 2.
 12.30: Calculate the magnitude of the forces on the two ropes in Figure 2....
 12.31: A 50kg wagon is pulled to the right by a force F1 making an angle ...
 12.32: Let v, w, and u be the vectors in R3. Which of the following is a s...
 12.33: In Exercises 3336, let v = 1, 2, 4, u = 6, 1, 2, and w = 1, 0, 3. C...
 12.34: In Exercises 3336, let v = 1, 2, 4, u = 6, 1, 2, and w = 1, 0, 3. C...
 12.35: In Exercises 3336, let v = 1, 2, 4, u = 6, 1, 2, and w = 1, 0, 3. C...
 12.36: In Exercises 3336, let v = 1, 2, 4, u = 6, 1, 2, and w = 1, 0, 3. C...
 12.37: Use the cross product to find the area of the triangle whose vertic...
 12.38: Calculate v w if v = 2, v w = 3, and the angle between v and w is 6...
 12.39: Show that if the vectors v, w are orthogonal, then v + w2 = v2 + w2...
 12.40: Find the angle between v and w if v + w=v=w. 41. F
 12.41: Find e 4f, assuming that e and f are unit vectors such that e + f =...
 12.42: Find the area of the parallelogram spanned by vectors v and w such ...
 12.43: Show that the equation 1, 2, 3 v = 1, 2, a has no solution for a = 1.
 12.44: Prove with a diagram the following: If e is a unit vector orthogona...
 12.45: Use the identity u (v w) = (u w) v (u v) w to prove that u (v w) + ...
 12.46: Find an equation of the plane through (1, 3, 5) with normal vector ...
 12.47: Write the equation of the plane P with vector equation 1, 4, 3x, y,...
 12.48: Find all the planes parallel to the plane passing through the point...
 12.49: Find the plane through P = (4, 1, 9) containing the line r(t) = 1, ...
 12.50: Find the intersection of the line r(t) = 3t + 2, 1, 7t and the plan...
 12.51: Find the intersection of the line r(t) = 3t + 2, 1, 7t and the plan...
 12.52: Find the intersection of the planes x + y + z = 1 and 3x 2y + z = 5.
 12.53: In Exercises 5358, determine the type of the quadric surface. 53. x...
 12.54: In Exercises 5358, determine the type of the quadric surface.x 3 2 ...
 12.55: In Exercises 5358, determine the type of the quadric surface.x 3 2 ...
 12.56: In Exercises 5358, determine the type of the quadric surface.x 3 2 ...
 12.57: In Exercises 5358, determine the type of the quadric surface.x 3 2 ...
 12.58: In Exercises 5358, determine the type of the quadric surface.x 3 2 ...
 12.59: Determine the type of the quadric surface ax2 + by2 z2 = 1 if: (a) ...
 12.60: Describe the traces of the surface x 2 2 y2 + z 2 2 = 1 in the thre...
 12.61: Convert (x, y, z) = (3, 4, 1) from rectangular to cylindrical and s...
 12.62: Convert (r, , z) = 3, 6 , 4 from cylindrical to spherical coordinates.
 12.63: Convert the point ( , , ) = 3, 6 , 3 from spherical to cylindrical ...
 12.64: Describe the set of all points P = (x, y,z) satisfying x2 + y2 4 in...
 12.65: Sketch the graph of the cylindrical equation z = 2r cos and write t...
 12.66: Write the surface x2 + y2 z2 = 2 (x + y) as an equation r = f (,z) ...
 12.67: Show that the cylindrical equation r2(1 2 sin2 ) + z2 = 1 is a hype...
 12.68: Sketch the graph of the spherical equation = 2 cos sin and write th...
 12.69: Describe how the surface with spherical equation 2(1 + A cos2 ) = 1...
 12.70: Show that the spherical equation cot = 2 cos + sin defines a plane ...
 12.71: Let c be a scalar, a and b be vectors, and X = x, y,z. Show that th...
Solutions for Chapter 12: VECTOR GEOMETRY
Full solutions for Calculus: Early Transcendentals  3rd Edition
ISBN: 9781464114885
Solutions for Chapter 12: VECTOR GEOMETRY
Get Full SolutionsSince 71 problems in chapter 12: VECTOR GEOMETRY have been answered, more than 41798 students have viewed full stepbystep solutions from this chapter. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9781464114885. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals , edition: 3. Chapter 12: VECTOR GEOMETRY includes 71 full stepbystep solutions.

Augmented matrix
A matrix that represents a system of equations.

Average velocity
The change in position divided by the change in time.

Base
See Exponential function, Logarithmic function, nth power of a.

Circular functions
Trigonometric functions when applied to real numbers are circular functions

Common logarithm
A logarithm with base 10.

Commutative properties
a + b = b + a ab = ba

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

Continuous at x = a
lim x:a x a ƒ(x) = ƒ(a)

Coordinate(s) of a point
The number associated with a point on a number line, or the ordered pair associated with a point in the Cartesian coordinate plane, or the ordered triple associated with a point in the Cartesian threedimensional space

Permutation
An arrangement of elements of a set, in which order is important.

Phase shift
See Sinusoid.

Plane in Cartesian space
The graph of Ax + By + Cz + D = 0, where A, B, and C are not all zero.

Reflexive property of equality
a = a

Sample survey
A process for gathering data from a subset of a population, usually through direct questioning.

Sequence
See Finite sequence, Infinite sequence.

Series
A finite or infinite sum of terms.

Trichotomy property
For real numbers a and b, exactly one of the following is true: a < b, a = b , or a > b.

Vector
An ordered pair <a, b> of real numbers in the plane, or an ordered triple <a, b, c> of real numbers in space. A vector has both magnitude and direction.

Viewing window
The rectangular portion of the coordinate plane specified by the dimensions [Xmin, Xmax] by [Ymin, Ymax].

Weights
See Weighted mean.