Stewart Section 12.4: The Cross Product

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Stewart Section 12.4: The Cross Product 0/52
1 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a} = \langle 2, 3, 0 \rangle\), \(\mathbf{b} = \langle 1, 0, 5 \rangle\)
2 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a} = \langle 4, 3, -2 \rangle\), \(\mathbf{b} = \langle 2, -1, 1 \rangle\)
3 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a} = 2 \mathbf{j} - 4 \mathbf{k}\), \(\mathbf{b} = -\mathbf{i} + 3 \mathbf{j} + \mathbf{k}\)
4 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a} = 3 \mathbf{i} + 3 \mathbf{j} - 3 \mathbf{k}\), \(\mathbf{b} = 3 \mathbf{i} - 3 \mathbf{j} + 3 \mathbf{k}\)
5 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a} = \dfrac{1}{2} \mathbf{i} + \dfrac{1}{3} \mathbf{j} + \dfrac{1}{4} \mathbf{k}\), \(\mathbf{b} = \mathbf{i} + 2 \mathbf{j} - 3 \mathbf{k}\)
6 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a} = t \mathbf{i} + \cos t \mathbf{j} + \sin t \mathbf{k}\), \(\mathbf{b} = \mathbf{i} - \sin t \mathbf{j} + \cos t \mathbf{k}\)
7 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that it is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). \(\mathbf{a} = \langle t, 1, \dfrac{1}{t} \rangle\), \(\mathbf{b} = \langle t^2, t^2, 1 \rangle\)
8 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
If \(\mathbf{a} = \mathbf{i} - 2 \mathbf{k}\) and \(\mathbf{b} = \mathbf{j} + \mathbf{k}\), find \(\mathbf{a} \times \mathbf{b}\). Sketch \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{a} \times \mathbf{b}\) as vectors starting at the origin.
9 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find the vector, not with determinants, but by using properties of cross products. \((\mathbf{i} \times \mathbf{j}) \times \mathbf{k}\)
10 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find the vector, not with determinants, but by using properties of cross products. \(\mathbf{k} \times (\mathbf{i} - 2 \mathbf{j})\)
11 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the vector, not with determinants, but by using properties of cross products. \((\mathbf{j} - \mathbf{k}) \times (\mathbf{k} - \mathbf{i})\)
12 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find the vector, not with determinants, but by using properties of cross products. \((\mathbf{i} + \mathbf{j}) \times (\mathbf{i} - \mathbf{j})\)
13 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar.
(a) \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\)
(b) \(\mathbf{a} \times (\mathbf{b} \cdot \mathbf{c})\)
(c) \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})\)
(d) \(\mathbf{a} \cdot (\mathbf{b} \cdot \mathbf{c})\)
(e) \((\mathbf{a} \cdot \mathbf{b}) \times (\mathbf{c} \cdot \mathbf{d})\)
(f) \((\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d})\)

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14 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find \(|\mathbf{u} \times \mathbf{v}|\) and determine whether \(\mathbf{u} \times \mathbf{v}\) is directed into the page or out of the page.
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15 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
Find \(|\mathbf{u} \times \mathbf{v}|\) and determine whether \(\mathbf{u} \times \mathbf{v}\) is directed into the page or out of the page.
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16 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
The figure shows a vector \(\mathbf{a}\) in the \(x y\)-plane and a vector \(\mathbf{b}\) in the direction of \(\mathbf{k}\). Their lengths are \(|\mathbf{a}| = 3\) and \(|\mathbf{b}| = 2\).
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(a) Find \(|\mathbf{a} \times \mathbf{b}|\).
(b) Use the right-hand rule to decide whether the components of \(\mathbf{a} \times \mathbf{b}\) are positive, negative, or 0.

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17 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 2
If \(\mathbf{a} = \langle 2, -1, 3 \rangle\) and \(\mathbf{b} = \langle 4, 2, 1 \rangle\), find \(\mathbf{a} \times \mathbf{b}\) and \(\mathbf{b} \times \mathbf{a}\).
18 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
If \(\mathbf{a} = \langle 1, 0, 1 \rangle\), \(\mathbf{b} = \langle 2, 1, -1 \rangle\), and \(\mathbf{c} = \langle 0, 1, 3 \rangle\), show that \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\).
19 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find two unit vectors orthogonal to both \(\langle 3, 2, 1 \rangle\) and \(\langle -1, 1, 0 \rangle\).
20 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find two unit vectors orthogonal to both \(\mathbf{i} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\).
21 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Show that \(\mathbf{0} \times \mathbf{a} = \mathbf{0} = \mathbf{a} \times \mathbf{0}\) for any vector \(\mathbf{a}\) in \(V_3\).
22 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Show that \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{b} = 0\) for all vectors \(\mathbf{a}\) and \(\mathbf{b}\) in \(V_3\).
23 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Prove the property of cross products (Theorem 11). Property 1: \(\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}\)
24 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Prove the property of cross products (Theorem 11). Property 2: \((c \mathbf{a}) \times \mathbf{b} = c(\mathbf{a} \times \mathbf{b}) = \mathbf{a} \times (c \mathbf{b})\)
25 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Prove the property of cross products (Theorem 11). Property 3: \(\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}\)
26 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Prove the property of cross products (Theorem 11). Property 4: \((\mathbf{a} + \mathbf{b}) \times \mathbf{c} = \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c}\)
27 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the area of the parallelogram with vertices \(A(-3, 0)\), \(B(-1, 3)\), \(C(5, 2)\), and \(D(3, -1)\).
28 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the area of the parallelogram with vertices \(P(1, 0, 2)\), \(Q(3, 3, 3)\), \(R(7, 5, 8)\), and \(S(5, 2, 7)\).
29 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
(a) Find a nonzero vector orthogonal to the plane through the points \(P\), \(Q\), and \(R\), and (b) find the area of triangle \(P Q R\). \(P(1, 0, 1)\), \(Q(-2, 1, 3)\), \(R(4, 2, 5)\)
30 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
(a) Find a nonzero vector orthogonal to the plane through the points \(P\), \(Q\), and \(R\), and (b) find the area of triangle \(P Q R\). \(P(0, 0, -3)\), \(Q(4, 2, 0)\), \(R(3, 3, 1)\)
31 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
(a) Find a nonzero vector orthogonal to the plane through the points \(P\), \(Q\), and \(R\), and (b) find the area of triangle \(P Q R\). \(P(0, -2, 0)\), \(Q(4, 1, -2)\), \(R(5, 3, 1)\)
32 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
(a) Find a nonzero vector orthogonal to the plane through the points \(P\), \(Q\), and \(R\), and (b) find the area of triangle \(P Q R\). \(P(2, -3, 4)\), \(Q(-1, -2, 2)\), \(R(3, 1, -3)\)
33 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the volume of the parallelepiped determined by the vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\). \(\mathbf{a} = \langle 1, 2, 3 \rangle\), \(\mathbf{b} = \langle -1, 1, 2 \rangle\), \(\mathbf{c} = \langle 2, 1, 4 \rangle\)
34 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the volume of the parallelepiped determined by the vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\). \(\mathbf{a} = \mathbf{i} + \mathbf{j}\), \(\mathbf{b} = \mathbf{i} + \mathbf{k}\), \(\mathbf{c} = \mathbf{i} + \mathbf{j} + \mathbf{k}\)
35 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the volume of the parallelepiped with adjacent edges \(P Q\), \(P R\), and \(P S\). \(P(-2, 1, 0)\), \(Q(2, 3, 2)\), \(R(1, 4, -1)\), \(S(3, 6, 1)\)
36 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Find the volume of the parallelepiped with adjacent edges \(P Q\), \(P R\), and \(P S\). \(P(3, 0, 1)\), \(Q(-1, 2, 5)\), \(R(5, 1, -1)\), \(S(0, 4, 2)\)
37 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Use the scalar triple product to verify that the vectors \(\mathbf{u} = \mathbf{i} + 5 \mathbf{j} - 2 \mathbf{k}\), \(\mathbf{v} = 3 \mathbf{i} - \mathbf{j}\), and \(\mathbf{w} = 5 \mathbf{i} + 9 \mathbf{j} - 4 \mathbf{k}\) are coplanar.
38 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
Use the scalar triple product to determine whether the points \(A(1, 3, 2)\), \(B(3, -1, 6)\), \(C(5, 2, 0)\), and \(D(3, 6, -4)\) lie in the same plane.
39 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about \(P\).
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40 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
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(a) A horizontal force of 20 lb is applied to the handle of a gearshift lever as shown. Find the magnitude of the torque about the pivot point \(P\).
(b) Find the magnitude of the torque about \(P\) if the same force is applied at the elbow \(Q\) of the lever.

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41 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 3
A wrench 30 cm long lies along the positive \(y\)-axis and grips a bolt at the origin. A force is applied in the direction \(\langle 0, 3, -4 \rangle\) at the end of the wrench. Find the magnitude of the force needed to supply 100 N\(\cdot\)m of torque to the bolt.
42 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Let \(\mathbf{v} = 5 \mathbf{j}\) and let \(\mathbf{u}\) be a vector with length 3 that starts at the origin and rotates in the \(x y\)-plane. Find the maximum and minimum values of the length of the vector \(\mathbf{u} \times \mathbf{v}\). In what direction does \(\mathbf{u} \times \mathbf{v}\) point?
43 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
If \(\mathbf{a} \cdot \mathbf{b} = \sqrt{3}\) and \(\mathbf{a} \times \mathbf{b} = \langle 1, 2, 2 \rangle\), find the angle between \(\mathbf{a}\) and \(\mathbf{b}\).
44 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
(a) Find all vectors \(\mathbf{v}\) such that \(\langle 1, 2, 1 \rangle \times \mathbf{v} = \langle 3, 1, -5 \rangle\)
(b) Explain why there is no vector \(\mathbf{v}\) such that \(\langle 1, 2, 1 \rangle \times \mathbf{v} = \langle 3, 1, 5 \rangle\)

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45 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
(a) Let \(P\) be a point not on the line \(L\) that passes through the points \(Q\) and \(R\). Show that the distance \(d\) from the point \(P\) to the line \(L\) is \(d = \dfrac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a}|}\) where \(\mathbf{a} = \overrightarrow{Q R}\) and \(\mathbf{b} = \overrightarrow{Q P}\).
(b) Use the formula in part (a) to find the distance from the point \(P(1, 1, 1)\) to the line through \(Q(0, 6, 8)\) and \(R(-1, 4, 7)\).

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46 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
(a) Let \(P\) be a point not on the plane that passes through the points \(Q\), \(R\), and \(S\). Show that the distance \(d\) from \(P\) to the plane is \(d = \dfrac{|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|}{|\mathbf{a} \times \mathbf{b}|}\) where \(\mathbf{a} = \overrightarrow{Q R}\), \(\mathbf{b} = \overrightarrow{Q S}\), and \(\mathbf{c} = \overrightarrow{Q P}\).
(b) Use the formula in part (a) to find the distance from the point \(P(2, 1, 4)\) to the plane through the points \(Q(1, 0, 0)\), \(R(0, 2, 0)\), and \(S(0, 0, 3)\).

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47 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Show that \(|\mathbf{a} \times \mathbf{b}|^2 = |\mathbf{a}|^2 |\mathbf{b}|^2 - (\mathbf{a} \cdot \mathbf{b})^2\).
48 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
If \(\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}\), show that \(\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a}\)
49 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 4
Prove that \((\mathbf{a} - \mathbf{b}) \times (\mathbf{a} + \mathbf{b}) = 2(\mathbf{a} \times \mathbf{b})\).
50 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5
Prove Property 6 of cross products, that is, \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}\)
51 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5
Use Exercise 50 to prove that \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0}\)
52 Parametric Equations, Polar Coordinates, and Vector-Valued Functions · Level 5
Prove that \((\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = \begin{pmatrix} \mathbf{a} \cdot \mathbf{c} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{a} \cdot \mathbf{d} & \mathbf{b} \cdot \mathbf{d} \end{pmatrix}\)

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