# 5.4E: Excersies

- Page ID
- 25931

## 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) \(\displaystyle u=⟨2,0,0⟩, v=⟨2,2,0⟩\)

**Answer**-
\(\displaystyle a. u×v=⟨0,0,4⟩;\)

b.

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

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

**Answer**-
a. \(\displaystyle a. u×v=⟨6,−4,2⟩;\)

b.

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

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

**Answer**-
\(\displaystyle −2j−4k\)

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

**Answer**-
\(\displaystyle w=−\frac{1}{3\sqrt{6}}i−\frac{7}{3\sqrt{6}}j−\frac{2}{3\sqrt{6}}k\)

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 \(\displaystyle C(0,0,5)\)

**Answer**-
\(\displaystyle w=−\frac{4}{\sqrt{21}}i−\frac{2}{\sqrt{21}}j−\frac{1}{\sqrt{21}}k\)

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 \(\displaystyle v=4i−2j+αk.\)

**Answer**-
\(\displaystyle α=10\)

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 \(\displaystyle \begin{bmatrix}i&j&K\\1&&−1&7\\2&0&3\end{bmatrix}\).

**Answer**-
\(\displaystyle −3i+11j+2k\)

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

**Answer**-
\(\displaystyle w=⟨−1,e^t,−e^{−t}⟩\)

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 \(\displaystyle c=i+j+k.\)

**Answer**-
\(\displaystyle −26i+17j+9k\)

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.

**Answer**-
\(\displaystyle 72°\)

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 \(\displaystyle v=⟨0,2,1⟩\).

**Answer**-
\(\displaystyle 7\)

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**.

**Answer**-
\(\displaystyle a. 5\sqrt{6}; b. \frac{5\sqrt{6}}{2}; c. \frac{5\sqrt{6}}{\sqrt{59}}\)

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 \(\displaystyle w=i+k\)

**Answer**-
\(\displaystyle a. 2; b. 2\)

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 \(\displaystyle w=⟨4,2,7⟩.\)

**Answer**-
\(\displaystyle v⋅(u×w)=−1, w⋅(u×v)=1\)

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.

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

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 \(\displaystyle 3\) units^{3}^{.}

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

**Answer**-
\(\displaystyle a. α=1; b. h=1,\)

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)?

**Answer**-
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.

**Answer**-
\(\displaystyle −k\)

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 \(\displaystyle P(1,2,4).\)

**Answer**-
\(\displaystyle ⟨0,±4\sqrt{5},2\sqrt{5}⟩\)

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 \(\displaystyle ⟨1,1,1⟩×w=⟨−1,−1,2⟩.\)

**Answer**-
\(\displaystyle w=⟨w_3−1,w_3+1,w_3⟩,\) 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.

**Answer**-
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.

**Answer**-
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 \(\displaystyle B=0.3j\).

**Answer**-
\(\displaystyle F=4.8×10^{−15}kN\)

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.

**Answer**-
\(\displaystyle a. B(t)=⟨\frac{2sint}{\sqrt{5}},−\frac{2cost}{\sqrt{5}},\frac{1}{\sqrt{5}}⟩;\)

b.

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.)