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# 5.4E: Excersies

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## Exercise $$\PageIndex{1}$$

For the following exercises, the vectors $$\displaystyle u$$ and $$\displaystyle v$$ are given.

a. Find the cross product $$\displaystyle u×v$$ of the vectors $$\displaystyle u$$ and $$\displaystyle v$$. Express the answer in component form.

b. Sketch the vectors $$\displaystyle u,v,$$ and $$\displaystyle u×v.$$

1) $$\displaystyle u=⟨2,0,0⟩, v=⟨2,2,0⟩$$

$$\displaystyle a. u×v=⟨0,0,4⟩;$$

b.

2) $$\displaystyle u=⟨3,2,−1⟩, v=⟨1,1,0⟩$$

3) $$\displaystyle u=2i+3j, v=j+2k$$

a. $$\displaystyle a. u×v=⟨6,−4,2⟩;$$

b.

4) $$\displaystyle u=2j+3k, v=3i+k$$

5) Simplify $$\displaystyle (i×i−2i×j−4i×k+3j×k)×i.$$

$$\displaystyle −2j−4k$$

6) Simplify $$\displaystyle j×(k×j+2j×i−3j×j+5i×k).$$

## Exercise $$\PageIndex{2}$$

In the following exercises, vectors $$\displaystyle u$$ and $$\displaystyle v$$ are given. Find unit vector $$\displaystyle w$$ in the direction of the cross product vector $$\displaystyle u×v.$$ Express your answer using standard unit vectors.

7) $$\displaystyle u=⟨3,−1,2⟩, v=⟨−2,0,1⟩$$

$$\displaystyle w=−\frac{1}{3\sqrt{6}}i−\frac{7}{3\sqrt{6}}j−\frac{2}{3\sqrt{6}}k$$

8) $$\displaystyle u=⟨2,6,1⟩, v=⟨3,0,1⟩$$

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

$$\displaystyle w=−\frac{4}{\sqrt{21}}i−\frac{2}{\sqrt{21}}j−\frac{1}{\sqrt{21}}k$$

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

## Exercise $$\PageIndex{3}$$

11) Determine the real number $$\displaystyle α$$ such that $$\displaystyle u×v$$ and $$\displaystyle i$$ are orthogonal, where $$\displaystyle u=3i+j−5k$$ and $$\displaystyle v=4i−2j+αk.$$

$$\displaystyle α=10$$

12) Show that $$\displaystyle u×v$$ and $$\displaystyle 2i−14j+2k$$ cannot be orthogonal for any α real number, where $$\displaystyle u=i+7j−k$$ and $$\displaystyle v=αi+5j+k$$.

13) Show that $$\displaystyle u×v$$ is orthogonal to $$\displaystyle u+v$$ and $$\displaystyle u−v$$, where $$\displaystyle u$$ and $$\displaystyle v$$ are nonzero vectors.

14) Show that $$\displaystyle v×u$$ is orthogonal to $$\displaystyle (u⋅v)(u+v)+u$$, where $$\displaystyle u$$ and $$\displaystyle v$$ are nonzero vectors.

15) Calculate the determinant $$\displaystyle \begin{bmatrix}i&j&K\\1&&−1&7\\2&0&3\end{bmatrix}$$.

$$\displaystyle −3i+11j+2k$$

16) Calculate the determinant $$\displaystyle \begin{bmatrix}i&j&K\\0&3&−4\\1&6&−1\end{bmatrix}$$.

## Exercise $$\PageIndex{4}$$

For the following exercises, the vectors $$\displaystyle u$$ and $$\displaystyle v$$ are given. Use determinant notation to find vector $$\displaystyle w$$ orthogonal to vectors $$\displaystyle u$$ and $$\displaystyle v$$.

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

$$\displaystyle w=⟨−1,e^t,−e^{−t}⟩$$

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

19) Find vector $$\displaystyle (a−2b)×c,$$ where $$\displaystyle a=\begin{bmatrix}i&j&k\\2&−1&5\\0&1&8\end{bmatrix}, b=\begin{bmatrix}i&j&K\\0&1&1\\2&−1&−2\end{bmatrix},$$ and $$\displaystyle c=i+j+k.$$

$$\displaystyle −26i+17j+9k$$

20) Find vector $$\displaystyle c×(a+3b),$$ where $$\displaystyle a=\begin{bmatrix}i&j&K\\5&0&9\\0&1&0\end{bmatrix}, b=\begin{bmatrix}i&j&k\\0&−1&1\\7&1&−1\end{bmatrix},$$ and $$\displaystyle c=i−k.$$

## Exercise $$\PageIndex{5}$$

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

$$\displaystyle 72°$$

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

## Exercise $$\PageIndex{6}$$

23) Use the sine and cosine of the angle between two nonzero vectors $$\displaystyle u$$ and $$\displaystyle v$$ to prove Lagrange’s identity: $$\displaystyle ‖u×v‖^2=‖u‖^2‖v‖^2−(u⋅v)^2$$.

24) Verify Lagrange’s identity $$\displaystyle ‖u×v‖^2=‖u‖^2‖v‖^2−(u⋅v)^2$$ for vectors $$\displaystyle u=−i+j−2k$$ and $$\displaystyle v=2i−j.$$

25) Nonzero vectors $$\displaystyle u$$ and $$\displaystyle v$$ are called collinear if there exists a nonzero scalar $$\displaystyle α$$ such that $$\displaystyle v=αu$$. Show that $$\displaystyle u$$ and $$\displaystyle v$$ are collinear if and only if $$\displaystyle u×v=0.$$

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

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

$$\displaystyle 7$$

28) Find the area of the parallelogram with adjacent sides $$\displaystyle u=i+j$$ and $$\displaystyle v=i+k.$$

## Exercise $$\PageIndex{7}$$

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

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

b. Find the area of triangle ABC.

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

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

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

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

b. Find the area of triangle ABC.

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

## Exercise $$\PageIndex{8}$$

In the following exercises, vectors $$\displaystyle u,v$$, and $$\displaystyle w$$ are given.

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

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

31) $$\displaystyle u=i+j, v=j+k,$$ and $$\displaystyle w=i+k$$

$$\displaystyle a. 2; b. 2$$

32) $$\displaystyle u=⟨−3,5,−1⟩, v=⟨0,2,−2⟩,$$ and $$\displaystyle w=⟨3,1,1⟩$$

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

$$\displaystyle v⋅(u×w)=−1, w⋅(u×v)=1$$

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

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

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

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

## Exercise $$\PageIndex{9}$$

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

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

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

$$\displaystyle a. α=1; b. h=1,$$

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

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

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

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

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

a. $$\displaystyle u×u=0$$

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

c. $$\displaystyle c(u×v)=(cu)×v=u×(cv)$$

d. $$\displaystyle u⋅(u×v)=0$$

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

a. $$\displaystyle u×u=0$$

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

c. $$\displaystyle c(u×v)=(cu)×v=u×(cv)$$

d. $$\displaystyle u⋅(u×v)=0$$

41) Nonzero vectors $$\displaystyle u,v,$$ and $$\displaystyle w$$ 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 $$\displaystyle α$$ and $$\displaystyle β$$ such that $$\displaystyle w=αu+βv$$. Otherwise, the vectors are called linearly independent. Show that $$\displaystyle u,v,$$ and $$\displaystyle w$$ are coplanar if and only if they are linear dependent.

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

a. Show that $$\displaystyle u,v,$$ and $$\displaystyle w$$ are coplanar by using their triple scalar product

b. Show that $$\displaystyle u,v,$$ and $$\displaystyle w$$ are coplanar, using the definition that there exist two nonzero real numbers $$\displaystyle α$$ and $$\displaystyle β$$ such that $$\displaystyle w=αu+βv.$$

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

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

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

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

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

$$\displaystyle \tilde{u}×\tilde{v}=\begin{bmatrix}i&j&k\\u_1&u_2&0\\v_1&v_2&0\end{bmatrix}$$.

Use this result to compute $$\displaystyle (icosθ+jsinθ)×(isinθ−jcosθ),$$ where $$\displaystyle θ$$ is a real number.

$$\displaystyle −k$$

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

a. Find the area of triangle $$\displaystyle P,Q,$$ and $$\displaystyle R.$$

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

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

$$\displaystyle ⟨0,±4\sqrt{5},2\sqrt{5}⟩$$

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

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

## Exercise $$\PageIndex{10}$$

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

a. Graph the function $$\displaystyle f$$.

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

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

51) Find all vectors $$\displaystyle w=⟨w_1,w_2,w_3⟩$$ that satisfy the equation $$\displaystyle ⟨1,1,1⟩×w=⟨−1,−1,2⟩.$$

$$\displaystyle w=⟨w_3−1,w_3+1,w_3⟩,$$ where $$\displaystyle w_3$$ is any real number

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

53) [T] A mechanic uses a 12-in. wrench to turn a bolt. The wrench makes a $$\displaystyle 30°$$ angle with the horizontal. If the mechanic applies a vertical force of $$\displaystyle 10$$ lb on the wrench handle, what is the magnitude of the torque at point $$\displaystyle P$$ (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 $$\displaystyle 40°$$ angle with the horizontal (see the following figure). Find the torque at point $$\displaystyle P$$. 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 $$\displaystyle ⟨0,1,−2⟩$$ and it produces a $$\displaystyle 100$$ N·m torque to the bolt located at the origin.

250 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 $$\displaystyle 35°$$ to produce a torque of $$\displaystyle 20$$ N·m?

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

$$\displaystyle F=4.8×10^{−15}kN$$

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

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

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

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

$$\displaystyle a. B(t)=⟨\frac{2sint}{\sqrt{5}},−\frac{2cost}{\sqrt{5}},\frac{1}{\sqrt{5}}⟩;$$
61) 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) $$\displaystyle A(8,0,0), B(8,18,0), C(0,18,8),$$ and $$\displaystyle D(0,0,8)$$ (see the following figure).
a. Find vector $$\displaystyle n=\vec{AB}×\vec{AD}$$ perpendicular to the surface of the solar panels. Express the answer using standard unit vectors.
b. Assume unit vector $$\displaystyle s=\frac{1}{\sqrt{3}}i+\frac{1}{\sqrt{3}}j+\frac{1}{\sqrt{3}}k$$ points toward the Sun at a particular time of the day and the flow of solar energy is $$\displaystyle F=900s$$ (in watts per square meter [$$\displaystyle W/m^2$$]). Find the predicted amount of electrical power the panel can produce, which is given by the dot product of vectors $$\displaystyle F$$ and $$\displaystyle n$$ (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 $$\displaystyle n$$ and $$\displaystyle s$$ and the angle of elevation are complementary.)