 12.3.1: Is the dot product of two vectors a scalar or a vector?
 12.3.2: What can you say about the angle between a and b if a b < 0?
 12.3.3: Which property of dot products allows us to conclude that if v is o...
 12.3.4: Which is the projection of v along v: (a) v or (b) ev?
 12.3.5: Let uv be the projection of u along v. Which of the following is ...
 12.3.6: Which of the following is equal to cos , where is the angle between...
 12.3.7: In Exercises 112, compute the dot product. k j
 12.3.8: In Exercises 112, compute the dot product. k k
 12.3.9: In Exercises 112, compute the dot product. (i + j) (j + k)
 12.3.10: In Exercises 112, compute the dot product. (3j + 2k) (i 4k)
 12.3.11: In Exercises 112, compute the dot product. (i + j + k) (3i + 2j 5k)
 12.3.12: In Exercises 112, compute the dot product. (k) (i 2j + 7k)
 12.3.13: In Exercises 1318, determine whether the two vectors are orthogonal...
 12.3.14: In Exercises 1318, determine whether the two vectors are orthogonal...
 12.3.15: In Exercises 1318, determine whether the two vectors are orthogonal...
 12.3.16: In Exercises 1318, determine whether the two vectors are orthogonal...
 12.3.17: In Exercises 1318, determine whether the two vectors are orthogonal...
 12.3.18: In Exercises 1318, determine whether the two vectors are orthogonal...
 12.3.19: In Exercises 1922, find the cosine of the angle between the vectors...
 12.3.20: In Exercises 1922, find the cosine of the angle between the vectors...
 12.3.21: In Exercises 1922, find the cosine of the angle between the vectors...
 12.3.22: In Exercises 1922, find the cosine of the angle between the vectors...
 12.3.23: In Exercises 2328, find the angle between the vectors. Use a calcul...
 12.3.24: In Exercises 2328, find the angle between the vectors. Use a calcul...
 12.3.25: In Exercises 2328, find the angle between the vectors. Use a calcul...
 12.3.26: In Exercises 2328, find the angle between the vectors. Use a calcul...
 12.3.27: In Exercises 2328, find the angle between the vectors. Use a calcul...
 12.3.28: In Exercises 2328, find the angle between the vectors. Use a calcul...
 12.3.29: Find all values of b for which the vectors are orthogonal. (a) b, 3...
 12.3.30: Find a vector that is orthogonal to 1, 2, 2.
 12.3.31: Find two vectors that are not multiples of each other and are both ...
 12.3.32: Find a vector that is orthogonal to v = 1, 2, 1 but not to w = 1, 0...
 12.3.33: Find v e where v = 3, e is a unit vector, and the angle between e a...
 12.3.34: Assume that v lies in the yzplane. Which of the following dot prod...
 12.3.35: In Exercises 3538, simplify the expression. 35. (v w) v + v w
 12.3.36: In Exercises 3538, simplify the expression.(v + w) (v + w) 2v w
 12.3.37: In Exercises 3538, simplify the expression.(v + w) v (v + w) w
 12.3.38: In Exercises 3538, simplify the expression.(v + w) v (v w) w
 12.3.39: In Exercises 3942, use the properties of the dot product to evaluat...
 12.3.40: In Exercises 3942, use the properties of the dot product to evaluat...
 12.3.41: In Exercises 3942, use the properties of the dot product to evaluat...
 12.3.42: In Exercises 3942, use the properties of the dot product to evaluat...
 12.3.43: Find the angle between v and w if v w = v w. 44.
 12.3.44: Find the angle between v and w if v w = 1 2 v w. 45.
 12.3.45: Assume that v = 3, w = 5, and the angle between v and w is = 3 . (a...
 12.3.46: Assume that v = 3, w = 5, and the angle between v and w is = 3 . (a...
 12.3.47: Show that if e and f are unit vectors such that e + f = 3 2 , then ...
 12.3.48: Find 2e 3f, assuming that e and f are unit vectors such that e + f ...
 12.3.49: Find the angle in the triangle in Figure 12.
 12.3.50: Find all three angles in the triangle in Figure 13.
 12.3.51: (a) Draw uv and vu for the vectors appearing as in Figure 14. (...
 12.3.52: Let u and v be two nonzero vectors. (a) Is it possible for the comp...
 12.3.53: In Exercises 5360, find the projection of u along v. 53. u = 2, 5, ...
 12.3.54: In Exercises 5360, find the projection of u along v.u = 2, 3, v = 1, 2
 12.3.55: In Exercises 5360, find the projection of u along v.u = 1, 2, 0, v ...
 12.3.56: In Exercises 5360, find the projection of u along v.u = 1, 1, 1, v ...
 12.3.57: In Exercises 5360, find the projection of u along v.u = 5i + 7j 4k,...
 12.3.58: In Exercises 5360, find the projection of u along v.u = i + 29k, v = j
 12.3.59: In Exercises 5360, find the projection of u along v.u = a, b, c, v = i
 12.3.60: In Exercises 5360, find the projection of u along v.u = a, a, b, v ...
 12.3.61: In Exercises 61 and 62, compute the component of u along v. u = 3, ...
 12.3.62: In Exercises 61 and 62, compute the component of u along v. u = 3, ...
 12.3.63: Find the length of OP in Figure 15.
 12.3.64: Find uv in Figure 15. x y u = 3, 5 v = 8, 2 uv P O FIGU
 12.3.65: In Exercises 6570, find the decomposition a = ab + ab with respec...
 12.3.66: In Exercises 6570, find the decomposition a = ab + ab with respec...
 12.3.67: In Exercises 6570, find the decomposition a = ab + ab with respec...
 12.3.68: In Exercises 6570, find the decomposition a = ab + ab with respec...
 12.3.69: In Exercises 6570, find the decomposition a = ab + ab with respec...
 12.3.70: In Exercises 6570, find the decomposition a = ab + ab with respec...
 12.3.71: Let e = cos ,sin . Show that e e = cos( ) for any two angles and .
 12.3.72: Let v and w be vectors in the plane. (a) UseTheorem 2 to explain wh...
 12.3.73: In Exercises 7376, refer to Figure 16. Find the angle between AB an...
 12.3.74: In Exercises 7376, refer to Figure 16. Find the angle between AB an...
 12.3.75: In Exercises 7376, refer to Figure 16. Calculate the projection of ...
 12.3.76: In Exercises 7376, refer to Figure 16. Calculate the projection of ...
 12.3.77: The methane molecule CH4 consists of a carbon molecule bonded to fo...
 12.3.78: Iron forms a crystal lattice where each central atom appears at the...
 12.3.79: Let v and w be nonzero vectors and set u = ev + ew. Use the dot pro...
 12.3.80: Let v, w, and a be nonzero vectors such that v a = w a. Is it true ...
 12.3.81: Calculate the force (in newtons) required to push a 40kg wagon up ...
 12.3.82: A force F is applied to each of two ropes (of negligible weight) at...
 12.3.83: A light beam travels along the ray determined by a unit vector L, s...
 12.3.84: Let P and Q be antipodal (opposite) points on a sphere of radius r ...
 12.3.85: Prove that v + w2 v w2 = 4v w. 86
 12.3.86: Use Exercise 85 to show that v and w are orthogonal if and only if ...
 12.3.87: Show that the two diagonals of a parallelogram are perpendicular if...
 12.3.88: Verify the Distributive Law: u (v + w) = u v + u w
 12.3.89: Verify that (v) w = (v w) for any scalar .
 12.3.90: Prove the Law of Cosines, c2 = a2 + b2 2ab cos , by referring to Fi...
 12.3.91: In this exercise, we prove the CauchySchwarz inequality: If v and w...
 12.3.92: Use (6) to prove the Triangle Inequality: v + wv+w Hint: First use ...
 12.3.93: This exercise gives another proof of the relation between the dot p...
 12.3.94: Let v = x, y and v = x cos + y sin , x sin + y cos Prove that the a...
 12.3.95: Let v be a nonzero vector. The angles , , between v and the unit ve...
 12.3.96: Find the direction cosines of v = 3, 6, 2
 12.3.97: The set of all points X = (x, y, z) equidistant from two points P, ...
 12.3.98: Sketch the plane consisting of all points X = (x, y, z) equidistant...
 12.3.99: Use Eq. (7) to find the equation of the plane consisting of all poi...
Solutions for Chapter 12.3: Dot Product and the Angle Between Two Vectors
Full solutions for Calculus: Early Transcendentals  3rd Edition
ISBN: 9781464114885
Solutions for Chapter 12.3: Dot Product and the Angle Between Two Vectors
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus: Early Transcendentals , edition: 3. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9781464114885. Chapter 12.3: Dot Product and the Angle Between Two Vectors includes 99 full stepbystep solutions. Since 99 problems in chapter 12.3: Dot Product and the Angle Between Two Vectors have been answered, more than 44628 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Bounded interval
An interval that has finite length (does not extend to ? or ?)

Combination
An arrangement of elements of a set, in which order is not important

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

Ellipse
The set of all points in the plane such that the sum of the distances from a pair of fixed points (the foci) is a constant

End behavior
The behavior of a graph of a function as.

equation of a hyperbola
(x  h)2 a2  (y  k)2 b2 = 1 or (y  k)2 a2  (x  h)2 b2 = 1

Fibonacci numbers
The terms of the Fibonacci sequence.

General form (of a line)
Ax + By + C = 0, where A and B are not both zero.

Geometric series
A series whose terms form a geometric sequence.

Line of travel
The path along which an object travels

NINT (ƒ(x), x, a, b)
A calculator approximation to ?ab ƒ(x)dx

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Random numbers
Numbers that can be used by researchers to simulate randomness in scientific studies (they are usually obtained from lengthy tables of decimal digits that have been generated by verifiably random natural phenomena).

Right triangle
A triangle with a 90° angle.

Solve a triangle
To find one or more unknown sides or angles of a triangle

Stretch of factor c
A transformation of a graph obtained by multiplying all the xcoordinates (horizontal stretch) by the constant 1/c, or all of the ycoordinates (vertical stretch) of the points by a constant c, c, > 1.

Supply curve
p = ƒ(x), where x represents production and p represents price

Symmetric matrix
A matrix A = [aij] with the property aij = aji for all i and j

Tangent line of ƒ at x = a
The line through (a, ƒ(a)) with slope ƒ'(a) provided ƒ'(a) exists.

Zoom out
A procedure of a graphing utility used to view more of the coordinate plane (used, for example, to find theend behavior of a function).