 12.4.1: What is the (1, 3) minor of the matrix 342 5 1 1 403 ?
 12.4.2: The angle between two unit vectors e and f is 6 . What is the lengt...
 12.4.3: What is u w, assuming that w u = 2, 2, 1?
 12.4.4: Find the cross product without using the formula: (a) 4, 8, 2 4, 8,...
 12.4.5: What are i j and i k?
 12.4.6: When is the cross product v w equal to zero?
 12.4.7: Which of the following are meaningful and which are not? Explain. (...
 12.4.8: Which of the following vectors are equal to j i? (a) i k (b) k (c) i j
 12.4.9: In Exercises 912, calculate v w. 9. v = 1, 2, 1, w = 3, 1, 1
 12.4.10: In Exercises 912, calculate v w.v = 2, 0, 0, w = 1, 0, 1
 12.4.11: In Exercises 912, calculate v w.v = 2 3 , 1, 1 2 , w = 4, 6, 3
 12.4.12: In Exercises 912, calculate v w.v = 2 3 , 1, 1 2 , w = 4, 6, 3
 12.4.13: In Exercises 1316, use the relations in Eq. (5) to calculate the cr...
 12.4.14: In Exercises 1316, use the relations in Eq. (5) to calculate the cr...
 12.4.15: In Exercises 1316, use the relations in Eq. (5) to calculate the cr...
 12.4.16: In Exercises 1316, use the relations in Eq. (5) to calculate the cr...
 12.4.17: In Exercises 1722, calculate the cross product assuming that u v = ...
 12.4.18: In Exercises 1722, calculate the cross product assuming that u v = ...
 12.4.19: In Exercises 1722, calculate the cross product assuming that u v = ...
 12.4.20: In Exercises 1722, calculate the cross product assuming that u v = ...
 12.4.21: In Exercises 1722, calculate the cross product assuming that u v = ...
 12.4.22: In Exercises 1722, calculate the cross product assuming that u v = ...
 12.4.23: Let v = a, b, c. Calculate v i, v j, and v k.
 12.4.24: Find v w, where v and w are vectors of length 3 in the xzplane, or...
 12.4.25: In Exercises 25 and 26, refer to Figure 17. Which of u and u is equ...
 12.4.26: In Exercises 25 and 26, refer to Figure 17. Which of the following ...
 12.4.27: Let v = 3, 0, 0 and w = 0, 1, 1. Determine u = v w using the geomet...
 12.4.28: What are the possible angles between two unit vectors e and f if e ...
 12.4.29: Show that if v and w lie in the yzplane, then v w is a multiple of i.
 12.4.30: Find the two unit vectors orthogonal to both a = 3, 1, 1 and b = 1,...
 12.4.31: Let e and e be unit vectors in R3 such that e e . Use the geometric...
 12.4.32: Calculate the force F on an electron (charge q = 1.6 1019 C) moving...
 12.4.33: An electron moving with velocity v in the plane experiences a force...
 12.4.34: Calculate the scalar triple product u (v w), where u = 1, 1, 0, v =...
 12.4.35: Verify identity (12) for vectors v = 3, 2, 2 and w = 4, 1, 2.
 12.4.36: Find the volume of the parallelepiped spanned by u, v, and w in Fig...
 12.4.37: Find the area of the parallelogram spanned by v and w in Figure 19.
 12.4.38: Calculate the volume of the parallelepiped spanned by u = 2, 2, 1, ...
 12.4.39: Sketch and compute the volume of the parallelepiped spanned by u = ...
 12.4.40: Sketch the parallelogram spanned by u = 1, 1, 1 and v = 0, 0, 4, an...
 12.4.41: Sketch the parallelogram spanned by u = 1, 1, 1 and v = 0, 0, 4, an...
 12.4.42: Find the area of the parallelogram determined by the vectors a, 0, ...
 12.4.43: Sketch the triangle with vertices at the origin O, P = (3, 3, 0), a...
 12.4.44: Use the cross product to find the area of the triangle with vertice...
 12.4.45: Use cross products to find the area of the triangle in the xyplane...
 12.4.46: Use cross products to find the area of the quadrilateral in the xyp...
 12.4.47: Check that the four points P (2, 4, 4), Q(3, 1, 6), R(2, 8, 0), and...
 12.4.48: Find three nonzero vectors a, b, and c such that a b = a c = 0 but ...
 12.4.49: In Exercises 4951, verify the identity using the formula for the cr...
 12.4.50: In Exercises 4951, verify the identity using the formula for the cr...
 12.4.51: In Exercises 4951, verify the identity using the formula for the cr...
 12.4.52: Use the geometric description in Theorem 1 to prove Theorem 2 (iii)...
 12.4.53: Verify the relations (5).
 12.4.54: Show that (i j) j = i (j j) Conclude that the Associative Law does ...
 12.4.55: The components of the cross product have a geometric interpretation...
 12.4.56: Formulate and prove analogs of the result in Exercise 55 for the i...
 12.4.57: Show that three points P , Q, R are collinear (lie on a line) if an...
 12.4.58: Use the result of Exercise 57 to determine whether the points P, Q,...
 12.4.59: Solve the equation 1, 1, 1 X = 1, 1, 0, where X = x, y,z. Note: The...
 12.4.60: Explain geometrically why 1, 1, 1 X = 1, 0, 0 has no solution, wher...
 12.4.61: Let X = x, y,z. Show that i X = v has a solution if and only if v i...
 12.4.62: Suppose that vectors u, v, and w are mutually orthogonalthat is, u ...
 12.4.63: In Exercises 6366, the torque about the origin O due to a force F a...
 12.4.64: In Exercises 6366, the torque about the origin O due to a force F a...
 12.4.65: In Exercises 6366, the torque about the origin O due to a force F a...
 12.4.66: In Exercises 6366, the torque about the origin O due to a force F a...
 12.4.67: Show that 3 3 determinants can be computed using the diagonal rule:...
 12.4.68: Use the diagonal rule to calculate 243 0 1 7 153
 12.4.69: Prove that v w = v u if and only if u = w + v for some scalar . Ass...
 12.4.70: Use Eq. (12) to prove the CauchySchwarz inequality: v wv w Show t...
 12.4.71: Show that if u, v, and w are nonzero vectors and (u v) w = 0, then ...
 12.4.72: Suppose that u, v, w are nonzero and (u v) w = u (v w) = 0 Show tha...
 12.4.73: Let a, b, c be nonzero vectors. Assume that b and c are not paralle...
 12.4.74: Use Exercise 73 to prove the identity (a b) c a (b c) = (a b)c (b c)a
 12.4.75: Show that if a, b are nonzero vectors such that a b, then there exi...
 12.4.76: Show that if a, b are nonzero vectors such that a b, then the set o...
 12.4.77: Assume that v and w lie in the first quadrant in R2 as in Figure 25...
 12.4.78: Consider the tetrahedron spanned by vectors a, b, and c as in Figur...
 12.4.79: In the notation of Exercise 78, suppose that a, b, c are mutually p...
Solutions for Chapter 12.4: The Cross Product
Full solutions for Calculus: Early Transcendentals  3rd Edition
ISBN: 9781464114885
Solutions for Chapter 12.4: The Cross Product
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus: Early Transcendentals , edition: 3. Since 79 problems in chapter 12.4: The Cross Product have been answered, more than 40733 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9781464114885. Chapter 12.4: The Cross Product includes 79 full stepbystep solutions.

Arccosecant function
See Inverse cosecant function.

Cardioid
A limaçon whose polar equation is r = a ± a sin ?, or r = a ± a cos ?, where a > 0.

Confounding variable
A third variable that affects either of two variables being studied, making inferences about causation unreliable

Focal length of a parabola
The directed distance from the vertex to the focus.

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.

Horizontal Line Test
A test for determining whether the inverse of a relation is a function.

Line of symmetry
A line over which a graph is the mirror image of itself

Multiplication principle of probability
If A and B are independent events, then P(A and B) = P(A) # P(B). If Adepends on B, then P(A and B) = P(AB) # P(B)

Multiplicative inverse of a complex number
The reciprocal of a + bi, or 1 a + bi = a a2 + b2 ba2 + b2 i

Normal distribution
A distribution of data shaped like the normal curve.

Paraboloid of revolution
A surface generated by rotating a parabola about its line of symmetry.

Parameter
See Parametric equations.

Polar axis
See Polar coordinate system.

Quadratic equation in x
An equation that can be written in the form ax 2 + bx + c = 01a ? 02

Reciprocal of a real number
See Multiplicative inverse of a real number.

Sinusoidal regression
A procedure for fitting a curve y = a sin (bx + c) + d to a set of data

Slope
Ratio change in y/change in x

Symmetric about the yaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

Unit vector
Vector of length 1.