# 12.4E: Exercises for The Cross Product


For exercises 1-4, the vectors $$\vecs{u}$$ and $$\vecs{v}$$ are given.

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 $$\vecs{u}\times\vecs{v}$$.

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

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

$$b.$$

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

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

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

$$b.$$

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

5) Simplify $$(\mathbf{\hat i}×\mathbf{\hat i}−2\mathbf{\hat i}×\mathbf{\hat j}−4\mathbf{\hat i}×\mathbf{\hat k}+3\mathbf{\hat j}×\mathbf{\hat k})×\mathbf{\hat i}.$$

$$−2\mathbf{\hat j}−4\mathbf{\hat k}$$

6) Simplify $$\mathbf{\hat j}×(\mathbf{\hat k}×\mathbf{\hat j}+2\mathbf{\hat j}×\mathbf{\hat i}−3\mathbf{\hat j}×\mathbf{\hat j}+5\mathbf{\hat i}×\mathbf{\hat k}).$$

In exercises 7-10, vectors $$\vecs{u}$$ and $$\vecs{v}$$ are given. Find unit vector $$\vecs{w}$$ in the direction of the cross product vector $$\vecs{u}×\vecs{v}.$$ Express your answer using standard unit vectors.

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

$$\vecs{w}=−\frac{\sqrt{6}}{18}\mathbf{\hat i}−\frac{7\sqrt{6}}{18}\mathbf{\hat j}−\frac{\sqrt{6}}{9}\mathbf{\hat k}$$

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

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

$$\vecs{w}=−\frac{4\sqrt{21}}{21}\mathbf{\hat i}−\frac{2\sqrt{21}}{21}\mathbf{\hat j}−\frac{\sqrt{21}}{21}\mathbf{\hat k}$$

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

11) 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 $$\vecs{v}=4\mathbf{\hat i}−2\mathbf{\hat j}+α\mathbf{\hat k}.$$

$$α=10$$

12) 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 $$\vecs{v}=α\mathbf{\hat i}+5\mathbf{\hat j}+\mathbf{\hat k}$$.

13) 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.

14) 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.

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

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

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

For exercises 17-18, the vectors $$\vecs{u}$$ and $$\vecs{v}$$ are given. Use determinant notation to find vector $$\vecs{w}$$ orthogonal to vectors $$\vecs{u}$$ and $$\vecs{v}$$.

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

$$\vecs{w}=⟨−1, e^t, −e^{−t}⟩$$

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

19) 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 $$\vecs{c}=\mathbf{\hat i}+\mathbf{\hat j}+\mathbf{\hat k}.$$

$$−26\mathbf{\hat i}+17\mathbf{\hat j}+9\mathbf{\hat k}$$

20) 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 $$\vecs{c}=\mathbf{\hat i}−\mathbf{\hat k}.$$

21) [T] Use the cross product $$\vecs{u}\times\vecs{v}$$ to find the acute angle between vectors $$\vecs{u}$$ and $$\vecs{v}$$, where $$\vecs{u}=\mathbf{\hat i}+2\mathbf{\hat j}$$ and $$\vecs{v}=\mathbf{\hat i}+\mathbf{\hat k}.$$ Express the answer in degrees rounded to the nearest integer.

$$72°$$

22) [T] Use the cross product $$\vecs{u}\times\vecs{v}$$ to find the obtuse angle between vectors $$\vecs{u}$$ and $$\vecs{v}$$, where $$\vecs{u}=−\mathbf{\hat i}+3\mathbf{\hat j}+\mathbf{\hat k}$$ and $$\vecs{v}=\mathbf{\hat i}−2\mathbf{\hat j}.$$ Express the answer in degrees rounded to the nearest integer.

23) Use the sine and cosine of the angle between two nonzero vectors $$\vecs u$$ and $$\vecs v$$ to prove Lagrange’s identity: $$\|\vecs{u}\times\vecs{v}\|^2=\|\vecs{u}\|^2\|\vecs{v}\|^2−(\vecs{u}⋅\vecs{v})^2$$.

24) Verify Lagrange’s identity $$\|\vecs{u}\times\vecs{v}\|^2=\|\vecs{u}\|^2\|\vecs{v}\|^2−(\vecs{u}⋅\vecs{v})^2$$ for vectors $$\vecs{u}=−\mathbf{\hat i}+\mathbf{\hat j}−2\mathbf{\hat k}$$ and $$\vecs{v}=2\mathbf{\hat i}−\mathbf{\hat j}.$$

25) Nonzero vectors $$\vecs{u}$$ and $$\vecs{v}$$ are called collinear if there exists a nonzero scalar $$α$$ such that $$\vecs{v}=α\vecs{u}$$. Show that $$\vecs{u}$$ and $$\vecs{v}$$ are collinear if and only if $$\vecs{u}\times\vecs{v}=0.$$

26) Nonzero vectors $$\vecs{u}$$ and $$\vecs{v}$$ are called collinear if there exists a nonzero scalar $$α$$ such that $$\vecs{v}=α\vecs{u}$$. Show that vectors $$\vecd{AB}$$ and $$\vecd{AC}$$ are collinear, where $$A(4,1,0), \, B(6,5,−2),$$ and $$C(5,3,−1).$$

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

$$7$$

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

29) 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 $$BC$$.

a. $$5\sqrt{6};$$ b. $$\frac{5\sqrt{6}}{2};$$ c. $$\frac{5\sqrt{6}}{\sqrt{59}} =\frac{5\sqrt{354}}{59}$$

30) 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 $$AC.$$

In exercises 31-32, vectors $$\vecs{u}, \, \vecs{v}$$, and $$\vecs{w}$$ are given.

a. Find the triple scalar product $$\vecs{u}⋅(\vecs{v}×\vecs{w}).$$

b. Find the volume of the parallelepiped with the adjacent edges $$\vecs{u},\,\vecs{v}$$, and $$\vecs{w}$$.

31) $$\quad \vecs{u}=\mathbf{\hat i}+\mathbf{\hat j}, \quad \vecs{v}=\mathbf{\hat j}+\mathbf{\hat k},$$ and $$\quad \vecs{w}=\mathbf{\hat i}+\mathbf{\hat k}$$

$$a. 2; \quad b. 2$$ units3

32) $$\quad \vecs{u}=⟨−3,5,−1⟩, \quad \vecs{v}=⟨0,2,−2⟩,$$ and $$\quad \vecs{w}=⟨3,1,1⟩$$

33) Calculate the triple scalar products $$\vecs{v}⋅(\vecs{u}×\vecs{w})$$ and $$\vecs{w}⋅(\vecs{u}×\vecs{v}),$$ where $$\vecs{u}=⟨1,1,1⟩, \vecs{v}=⟨7,6,9⟩,$$ and $$\vecs{w}=⟨4,2,7⟩.$$

$$\vecs{v}⋅(\vecs{u}×\vecs{w})=−1, \quad \vecs{w}⋅(\vecs{u}×\vecs{v})=1$$

34) Calculate the triple scalar products $$\vecs{w}⋅(\vecs{v}×\vecs{u})$$ and $$\vecs{u}⋅(\vecs{w}×\vecs{v}),$$ where $$\vecs{u}=⟨4,2,−1⟩, \vecs{v}=⟨2,5,−3⟩,$$ and $$\vecs{w}=⟨9,5,−10⟩.$$

35) Find vectors $$\vecs{a},\, \vecs{b}$$, and $$\vecs{c}$$ with a triple scalar product given by the determinant $$\begin{vmatrix}1&2&3\\0&2&5\\8&9&2\end{vmatrix}$$. Determine their triple scalar product.

$$\vecs{a}=⟨1,2,3⟩, \quad \vecs{b}=⟨0,2,5⟩, \quad \vecs{c}=⟨8,9,2⟩; \quad \vecs{a}⋅(\vecs{b}×\vecs{c})=−9$$

36) The triple scalar product of vectors $$\vecs{a},\,\vecs{b}$$, and $$\vecs{c}$$ is given by the determinant $$\begin{vmatrix}0&−2&1\\0&1&4\\1&−3&7\end{vmatrix}$$. Find vector $$\vecs{a}−\vecs{b}+\vecs{c}.$$

37) Consider the parallelepiped with edges $$OA,OB,$$ and $$OC$$, where $$A(2,1,0),B(1,2,0),$$ and $$C(0,1,α).$$

a. Find the real number $$α>0$$ such that the volume of the parallelepiped is $$3$$ units3.

b. For $$α=1,$$ find the height $$h$$ from vertex $$C$$ of the parallelepiped. Sketch the parallelepiped.

$$a. \, α=1; \quad b. \, h=1$$ unit,

38) Consider points $$A(α,0,0),B(0,β,0),$$ and $$C(0,0,γ)$$, with $$α, β$$, and $$γ$$ positive real numbers.

a. Determine the volume of the parallelepiped with adjacent sides $$\vecd{OA}, \vecd{OB},$$ and $$\vecd{OC}$$.

b. Find the volume of the tetrahedron with vertices $$O,A,B,$$ and $$C$$. (Hint: The volume of the tetrahedron is $$1/6$$ of the volume of the parallelepiped.)

c. Find the distance from the origin to the plane determined by $$A,B,$$ and $$C$$. Sketch the parallelepiped and tetrahedron.

39) Let $$u,v,$$ and $$w$$ be three-dimensional vectors and $$c$$ be a real number. Prove the following properties of the cross product.

a. $$\vecs u×\vecs u=\vecs 0$$

b. $$\vecs u×(\vecs v+\vecs w)=(\vecs u×\vecs v)+(\vecs u×\vecs w)$$

c. $$c(\vecs u×\vecs v)=(c\vecs u)×\vecs v=\vecs u×(c\vecs v)$$

d. $$\vecs u⋅(\vecs u×\vecs v)=\vecs 0$$

40) Show that vectors $$\vecs u=⟨1,0,−8⟩,\,\vecs v=⟨0,1,6⟩$$, and $$\vecs w=⟨−1,9,3⟩$$ satisfy the following properties of the cross product.

a. $$\vecs u×\vecs u=\vecs 0$$

b. $$\vecs u×(\vecs v+\vecs w)=(\vecs u×\vecs v)+(\vecs u×\vecs w)$$

c. $$c(\vecs u×\vecs v)=(c\vecs u)×\vecs v=\vecs u×(c\vecs v)$$

d. $$\vecs u⋅(\vecs u×\vecs v)=\vecs 0$$

41) Nonzero vectors $$\vecs u,\,\vecs v$$, and $$\vecs w$$ are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers $$α$$ and $$β$$ such that $$\vecs w=α\vecs u+β\vecs v$$. Otherwise, the vectors are called linearly independent. Show that $$\vecs u,\vecs v$$, and $$\vecs w$$ could be placed on the same plane if and only if they are linear dependent.

42) Consider vectors $$\vecs u=⟨1,4,−7⟩,\,\vecs v=⟨2,−1,4⟩,\,\vecs w=⟨0,−9,18⟩$$, and $$\vecs p=⟨0,−9,17⟩.$$

a. Show that $$\vecs u,\,\vecs v$$, and $$\vecs w$$ can be placed on the same plane by using their triple scalar product

b. Show that $$\vecs u,\,\vecs v$$, and $$\vecs w$$ can be placed on the same plane by using the definition that there exist two nonzero real numbers $$α$$ and $$β$$ such that $$w=αu+βv.$$

c. Show that $$\vecs u,\,\vecs v$$, and $$\vecs p$$ are linearly independent—that is, none of the vectors is a linear combination of the other two.

43) Consider points $$A(0,0,2), B(1,0,2), C(1,1,2),$$ and $$D(0,1,2).$$ Are vectors $$\vecd{AB}, \vecd{AC},$$ and $$\vecd{AD}$$ linearly dependent (that is, one of the vectors is a linear combination of the other two)?

Yes, $$\vecd{AD}=α\vecd{AB}+β\vecd{AC},$$ where $$α=−1$$ and $$β=1.$$

44) Show that vectors $$\mathbf{\hat i}+\mathbf{\hat j}, \mathbf{\hat i}−\mathbf{\hat j},$$ and $$\mathbf{\hat i}+\mathbf{\hat j}+\mathbf{\hat k}$$ are linearly independent—that is, there exist two nonzero real numbers $$α$$ and $$β$$ such that $$\mathbf{\hat i}+\mathbf{\hat j}+\mathbf{\hat k}=α(\mathbf{\hat i}+\mathbf{\hat j})+β(\mathbf{\hat i}−\mathbf{\hat j}).$$

45) Let $$\vecs u=⟨u_1,u_2⟩$$ and $$\vecs v=⟨v_1,v_2⟩$$ be two-dimensional vectors. The cross product of vectors $$\vecs u$$ and $$\vecs v$$ is not defined. However, if the vectors are regarded as the three-dimensional vectors $$\tilde{\vecs u}=⟨u_1,u_2,0⟩$$ and $$\tilde{\vecs v}=⟨v_1,v_2,0⟩$$, respectively, then, in this case, we can define the cross product of $$\tilde{\vecs u}$$ and $$\tilde{\vecs v}$$. In particular, in determinant notation, the cross product of $$\tilde{\vecs u}$$ and $$\tilde{\vecs v}$$ is given by

$$\tilde{\vecs u}×\tilde{\vecs v}=\begin{vmatrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\u_1&u_2&0\\v_1&v_2&0\end{vmatrix}$$.

Use this result to compute $$(\cos θ\,\mathbf{\hat i}+\sin θ\,\mathbf{\hat j})×(\sin θ\,\mathbf{\hat i}−\cos θ\,\mathbf{\hat j}),$$ where $$θ$$ is a real number.

$$−\mathbf{\hat k}$$

46) Consider points $$P(2,1), Q(4,2),$$ and $$R(1,2).$$

a. Find the area of triangle $$PQR$$.

b. Determine the distance from point $$R$$ to the line passing through $$P$$ and $$Q$$.

47) Determine a vector of magnitude $$10$$ perpendicular to the plane passing through the x-axis and point $$P(1,2,4).$$

$$⟨0,±4\sqrt{5},2\sqrt{5}⟩$$

48) Determine a unit vector perpendicular to the plane passing through the z-axis and point $$A(3,1,−2).$$

49) Consider $$\vecs u$$ and $$\vecs v$$ two three-dimensional vectors. If the magnitude of the cross product vector $$\vecs u×\vecs v$$ is $$k$$ times larger than the magnitude of vector $$\vecs u$$, show that the magnitude of $$\vecs v$$ is greater than or equal to $$k$$, where $$k$$ is a natural number.

50) [T] Assume that the magnitudes of two nonzero vectors $$\vecs u$$ and $$\vecs v$$ are known. The function $$f(θ)=‖\vecs u‖‖\vecs v‖\sin θ$$ defines the magnitude of the cross product vector $$\vecs u×\vecs v,$$ where $$θ∈[0,π]$$ is the angle between $$\vecs u$$ and $$\vecs v$$.

a. Graph the function $$f$$.

b. Find the absolute minimum and maximum of function $$f$$. Interpret the results.

c. If $$‖\vecs u‖=5$$ and $$‖\vecs v‖=2$$, find the angle between $$\vecs u$$ and $$\vecs v$$ if the magnitude of their cross product vector is equal to $$9$$.

51) Find all vectors $$\vecs w=⟨w_1,w_2,w_3⟩$$ that satisfy the equation $$⟨1,1,1⟩×\vecs w=⟨−1,−1,2⟩.$$ Hint: You should be able to write all components of $$\vecs w$$ in terms of one of the constants $$w_1,w_2,$$ or $$w_3$$.

Writing all components in terms of the constant $$w_3$$, one way to represent these vectors is: $$\vecs w=⟨w_3−1,w_3+1,w_3⟩,$$ where $$w_3$$ is any real number.
Note that we could use any parameter we wish here. We could set $$w_3 = a$$. Then $$\vecs w=⟨a−1,a+1,a⟩$$ would also represent these vectors.

52) Solve the equation $$\vecs w×⟨1,0,−1⟩=⟨3,0,3⟩,$$ where $$\vecs w=⟨w_1,w_2,w_3⟩$$ is a nonzero vector with a magnitude of $$3$$.

53) [T] A mechanic uses a 12-in. wrench to turn a bolt. The wrench makes a $$30°$$ angle with the horizontal. If the mechanic applies a vertical force of $$10$$ lb on the wrench handle, what is the magnitude of the torque at point $$P$$ (see the following figure)? Express the answer in foot-pounds rounded to two decimal places.

8.66 ft-lb

54) [T] A boy applies the brakes on a bicycle by applying a downward force of 20 lb on the pedal when the 6-in. crank makes a $$40°$$ angle with the horizontal (see the following figure). Find the torque at point $$P$$. Express your answer in foot-pounds rounded to two decimal places.

55) [T] Find the magnitude of the force that needs to be applied to the end of a 20-cm wrench located on the positive direction of the $$y$$-axis if the force is applied in the direction $$⟨0,1,−2⟩$$ and it produces a $$100$$ N·m torque to the bolt located at the origin.

$$250\sqrt{5}$$ N $$\approx 559$$ N

56) [T] What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of $$35°$$ to produce a torque of $$20$$ N·m?

57) [T] The force vector $$\vecs F$$ acting on a proton with an electric charge of $$1.6×10^{−19}\,C$$ (in coulombs) moving in a magnetic field $$\vecs B$$ where the velocity vector $$\vecs v$$ is given by $$\vecs F=1.6×10^{−19}(\vecs v×\vecs B)$$ (here, $$\vecs v$$ is expressed in meters per second, $$\vecs B$$ is in tesla [T], and $$\vecs F$$ is in newtons [N]). Find the force that acts on a proton that moves in the $$xy$$-plane at velocity $$\vecs v=10^5\mathbf{\hat i}+10^5\mathbf{\hat j}$$ (in meters per second) in a magnetic field given by $$\vecs B=0.3\mathbf{\hat j}$$.

$$\vecs F=4.8×10^{−15}\,kN$$

58) [T] The force vector $$\vecs F$$ acting on a proton with an electric charge of $$1.6×10^{−19}\,C$$ moving in a magnetic field $$\vecs B$$ where the velocity vector $$\vecs v$$ is given by $$\vecs F=1.6×10^{−19}(\vecs v×\vecs B)$$ (here, $$\vecs v$$ is expressed in meters per second, $$\vecs B$$ in $$T$$, and $$\vecs F$$ in $$N$$). If the magnitude of force $$\vecs F$$ acting on a proton is $$5.9×10^{−17}\,N$$ and the proton is moving at the speed of 300 m/sec in magnetic field $$\vecs B$$ of magnitude 2.4 T, find the angle between velocity vector $$\vecs v$$ of the proton and magnetic field $$\vecs B$$. Express the answer in degrees rounded to the nearest integer.

59) [T] Consider $$\vecs r(t)=⟨\cos t,\,\sin t,\,2t⟩$$ the position vector of a particle at time $$t∈[0,30]$$, where the components of $$\vecs r$$ are expressed in centimeters and time in seconds. Let $$\vecd{OP}$$ be the position vector of the particle after $$1$$ sec.

a. Determine unit vector $$\vecs B(t)$$ (called the binormal unit vector) that has the direction of cross product vector $$\vecs v(t)×\vecs a(t),$$ where $$\vecs v(t)$$ and $$\vecs a(t)$$ are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after $$t$$ seconds.

b. Use a CAS to visualize vectors $$\vecs v(1),\,\vecs a(1)$$, and $$\vecs B(1)$$ as vectors starting at point $$P$$ along with the path of the particle.

a. $$\vecs B(t)=⟨\frac{2\sqrt{5}\sin t}{5},−\frac{2\sqrt{5}\cos t}{5},\frac{\sqrt{5}}{5}⟩;$$

b.

60) 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.)