7.6e: Exercises - Cross Product

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A: Cross Products

Exercise $\PageIndex{A}$: Cross Products

1) Calculate the determinant $ \begin{vmatrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\1&−1&7\\2&0&3\end{vmatrix}$.

2) Calculate the determinant $ \begin{vmatrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\0&3&−4\\1&6&−1\end{vmatrix}$.

Given the vectors $\vecs{u}$ and $\vecs{v}$.

a. Find the cross product $\vecs{u}\times\vecs{v}$ of the vectors $\vecs{u}$ and $\vecs{v}$. Express the answer in component form.

b. Sketch the vectors $\vecs{u}, \, \vecs{v}$, and $\vecs{u}\times\vecs{v}$.

3) $\quad \vecs{u}=⟨2,0,0⟩, \quad \vecs{v}=⟨2,2,0⟩$

4) $\quad \vecs{u}=⟨3,2,−1⟩, \quad \vecs{v}=⟨1,1,0⟩$

5) $\quad \vecs{u}=2\mathbf{\hat i}+3\mathbf{\hat j}, \quad \vecs{v}=\mathbf{\hat j}+2\mathbf{\hat k}$

6) $\quad \vecs{u}=2\mathbf{\hat j}+3\mathbf{\hat k}, \quad \vecs{v}=3\mathbf{\hat i}+\mathbf{\hat k}$

7) Find vector $ (\vecs{a}−2\vecs{b})×\vecs{c},$ where $ \vecs{a}=\begin{vmatrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\2&−1&5\\0&1&8\end{vmatrix}, \vecs{b}=\begin{vmatrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\0&1&1\\2&−1&−2\end{vmatrix},$ and $\vecs{c}=\mathbf{\hat i}+\mathbf{\hat j}+\mathbf{\hat k}.$

8) Find vector $ \vecs{c}×(\vecs{a}+3\vecs{b}),$ where $ \vecs{a}=\begin{vmatrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\5&0&9\\0&1&0\end{vmatrix}, \vecs{b}=\begin{vmatrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\0&−1&1\\7&1&−1\end{vmatrix},$ and $\vecs{c}=\mathbf{\hat i}−\mathbf{\hat k}.$

Find vector $\vecs{w}$ orthogonal to vectors $\vecs{u}$ and $\vecs{v}$.

9) $\quad \vecs{u}=⟨−1, 0, e^t⟩, \quad \vecs{v}=⟨1, e^{−t}, 0⟩,$ where $t$ is a real number

10) $\quad \vecs{u}=⟨1, 0, x⟩, \quad \vecs{v}=⟨\frac{2}{x},1, 0⟩,$ where $x$ is a nonzero real number

Find unit vector $\vecs{w}$ in the direction of the cross product vector $\vecs{u}×\vecs{v}.$ Express your answer using standard unit vectors.

11) $\quad \vecs{u}=⟨3,−1,2⟩, \quad \vecs{v}=⟨−2,0,1⟩$

12) $\quad \vecs{u}=⟨2,6,1⟩, \quad \vecs{v}=⟨3,0,1⟩$

13) $\quad \vecs{u}=\vecd{AB}, \quad \vecs{v}=\vecd{AC},$ where $A(1,0,1),\, B(1,−1,3)$, and $C(0,0,5)$

14) $\quad \vecs{u}=\vecd{OP}, \quad \vecs{v}=\vecd{PQ},$ where $P(−1,1,0)$ and $Q(0,2,1)$

15) Determine the real number $α$ such that $\vecs{u}\times\vecs{v}$ and $\mathbf{\hat i}$ are orthogonal, where $\vecs{u}=3\mathbf{\hat i}+\mathbf{\hat j}−5\mathbf{\hat k}$ and $\vecs{v}=4\mathbf{\hat i}−2\mathbf{\hat j}+α\mathbf{\hat k}.$

16) Show that $\vecs{u}\times\vecs{v}$ and $ 2\mathbf{\hat i}−14\mathbf{\hat j}+2\mathbf{\hat k}$ cannot be orthogonal for any α real number, where $\vecs{u}=\mathbf{\hat i}+7\mathbf{\hat j}−\mathbf{\hat k}$ and $\vecs{v}=α\mathbf{\hat i}+5\mathbf{\hat j}+\mathbf{\hat k}$.

17) Show that $\vecs{u}\times\vecs{v}$ is orthogonal to $\vecs{u}+\vecs{v}$ and $\vecs{u}−\vecs{v}$, where $\vecs{u}$ and $\vecs{v}$ are nonzero vectors.

18) Show that $\vecs{v}\times\vecs{u}$ is orthogonal to $ (\vecs{u}⋅\vecs{v})(\vecs{u}+\vecs{v})+\vecs{u}$, where $\vecs{u}$ and $\vecs{v}$ are nonzero vectors.

21) Find the area of the parallelogram with adjacent sides $\vecs{u}=⟨3,2,0⟩$ and $\vecs{v}=⟨0,2,1⟩$.

22) Find the area of the parallelogram with adjacent sides $\vecs{u}=\mathbf{\hat i}+\mathbf{\hat j}$ and $\vecs{v}=\mathbf{\hat i}+\mathbf{\hat k}.$

23) Consider points $A(3,−1,2),\, B(2,1,5),$ and $C(1,−2,−2).$

a. Find the area of parallelogram $ABCD$ with adjacent sides $\vecd{AB}$ and $ \vecd{AC}$.

b. Find the area of triangle $ABC$.

c. Find the distance from point $A$ to line $BC$.

24) Consider points $A(2,−3,4),\, B(0,1,2),$ and $C(−1,2,0).$

a. Find the area of parallelogram $ABCD$ with adjacent sides $ \vecd{AB}$ and $ \vecd{AC}$.

b. Find the area of triangle $ABC$.

c. Find the distance from point $B$ to line $AC.$

$CNX_Calc_Figure_12_04_214.jfif$ 26) A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $ A(8,0,0), B(8,18,0), C(0,18,8),$ and $ D(0,0,8)$ (see the following figure).

a. Find vector $\vecs n=\vecd{AB}×\vecd{AD}$ perpendicular to the surface of the solar panels. Express the answer using standard unit vectors. Note that the magnitude of this vector should give us the area of rectangle $ABCD$.

b. Assume unit vector $\vecs s=\frac{1}{\sqrt{3}}\mathbf{\hat i}+\frac{1}{\sqrt{3}}\mathbf{\hat j}+\frac{1}{\sqrt{3}}\mathbf{\hat k}$ points toward the Sun at a particular time of the day and the flow of solar energy is $\vecs F=900\vecs s$ (in watts per square meter [$ W/m^2$]). Find the predicted amount of electrical power the panel can produce, which is given by the dot product of vectors $\vecs F$ and $\vecs n$ (expressed in watts).

c. Determine the angle of elevation of the Sun above the solar panel. Express the answer in degrees rounded to the nearest whole number. (Hint: The angle between vectors $\vecs n$ and $\vecs s$ and the angle of elevation are complementary.)

Answers 1-29:

1. $ −3\mathbf{\hat i}+11\mathbf{\hat j}+2\mathbf{\hat k}$

3. a. $\vecs{u}\times\vecs{v}=⟨0,0,4⟩;$

3. b.

$This figure is the first octant of the 3-dimensional coordinate system. On the x-axis there is a vector labeled “u.” In the x y-plane there is a vector labeled “v.” On the z-axis there is the vector labeled “u cross v.”$

5. a. $\ \vecs{u}\times\vecs{v}=⟨6,−4,2⟩;$

5. b.

$This figure is the first octant of the 3-dimensional coordinate system and shows three vectors. The first vector is labeled u and has components <2, 3, 0>. The second vector is labeled v and has components <0, 1, 2>.” The third vector is labeled u cross v and has components <6, -4, 2>.”$

7. $ −26\mathbf{\hat i}+17\mathbf{\hat j}+9\mathbf{\hat k}$ $ \quad $ 9. $\vecs{w}=⟨−1, e^t, −e^{−t}⟩$ $ \quad $
11. $\vecs{w}=−\frac{\sqrt{6}}{18}\mathbf{\hat i}−\frac{7\sqrt{6}}{18}\mathbf{\hat j}−\frac{\sqrt{6}}{9}\mathbf{\hat k}$
13. $\vecs{w}=−\frac{4\sqrt{21}}{21}\mathbf{\hat i}−\frac{2\sqrt{21}}{21}\mathbf{\hat j}−\frac{\sqrt{21}}{21}\mathbf{\hat k}$ $ \quad $ 15. $ α=10$ $ \quad $ 21. $ 7$ $ \quad $ 23. a. $5\sqrt{6};$ b. $\frac{5\sqrt{6}}{2};$ c. $\frac{5\sqrt{6}}{\sqrt{59}} =\frac{5\sqrt{354}}{59} $