
2.2E: Exercises for Vectors in Space


1) Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point $$A(2,3,5)$$ is the opposite vertex to the origin, then find

a. the coordinates of the other six vertices of the box and

b. the length of the diagonal of the box determined by the vertices $$O$$ and $$A$$.

a. $$(2,0,5),(2,0,0),(2,3,0),(0,3,0),(0,3,5),(0,0,5)$$   b. $$\sqrt{38}$$

2) Find the coordinates of point $$P$$ and determine its distance to the origin.

For exercises 3-6, describe and graph the set of points that satisfies the given equation.

3) $$(y−5)(z−6)=0$$

A union of two planes: $$y=5$$ (a plane parallel to the $$xz$$-plane) and $$z=6$$ (a plane parallel to the $$xy$$-plane)

4) $$(z−2)(z−5)=0$$

5) $$(y−1)^2+(z−1)^2=1$$

A cylinder of radius $$1$$ centered on the line $$y=1,z=1$$

6) $$(x−2)^2+(z−5)^2=4$$

7) Write the equation of the plane passing through point $$(1,1,1)$$ that is parallel to the $$xy$$-plane.

$$z=1$$

8) Write the equation of the plane passing through point $$(1,−3,2)$$ that is parallel to the $$xz$$-plane.

9) Find an equation of the plane passing through points $$(1,−3,−2), (0,3,−2),$$ and $$(1,0,−2).$$

$$z=−2$$

10) Find an equation of the plane passing through points $$(1,9,2), (1,3,6),$$ and $$(1,−7,8).$$

For exercises 11-14, find the equation of the sphere in standard form that satisfies the given conditions.

11) Center $$C(−1,7,4)$$ and radius $$4$$

$$(x+1)^2+(y−7)^2+(z−4)^2=16$$

12) Center $$C(−4,7,2)$$ and radius $$6$$

13) Diameter $$PQ,$$ where $$P(−1,5,7)$$ and $$Q(−5,2,9)$$

$$x+3)^2+(y−3.5)^2+(z−8)^2=\dfrac{29}{4}$$

14) Diameter $$PQ,$$ where $$P(−16,−3,9)$$ and $$Q(−2,3,5)$$

For exercises 15 and 16, find the center and radius of the sphere with an equation in general form that is given.

15) $$x^2+y^2+z^2−4z+3=0$$

Center $$C(0,0,2)$$ and radius $$1$$

16) $$x^2+y^2+z^2−6x+8y−10z+25=0$$

For exercises 17-20, express vector $$\vecd{PQ}$$ with the initial point at $$P$$ and the terminal point at $$Q$$

$$a.$$ in component form and

$$b.$$ by using standard unit vectors.

17) $$P(3,0,2)$$ and $$Q(−1,−1,4)$$

$$a. \vecd{PQ}=⟨−4,−1,2⟩$$
$$b. \vecd{PQ}=−4\hat{\mathbf i}−\hat{\mathbf j}+2\hat{\mathbf k}$$

18) $$P(0,10,5)$$ and $$Q(1,1,−3)$$

19) $$P(−2,5,−8)$$ and $$M(1,−7,4)$$, where $$M$$ is the midpoint of the line segment $$\overline{PQ}$$

$$a. \vecd{PQ}=⟨6,−24,24⟩$$
$$b. \vecd{PQ}=6\hat{\mathbf i}−24\hat{\mathbf j}+24\hat{\mathbf k}$$

20) $$Q(0,7,−6)$$ and $$M(−1,3,2)$$, where $$M$$ is the midpoint of the line segment $$\overline{PQ}$$

21) Find terminal point $$Q$$ of vector $$\vecd{PQ}=⟨7,−1,3⟩$$ with the initial point at $$P(−2,3,5).$$

$$Q(5,2,8)$$

22) Find initial point $$P$$ of vector $$\vecd{PQ}=⟨−9,1,2⟩$$ with the terminal point at $$Q(10,0,−1).$$

For exercises 23-26, use the given vectors $$\vecs a$$ and $$\vecs b$$ to find and express the vectors $$\vecs a+\vecs b, \,4\vecs a$$, and $$−5\vecs a+3\vecs b$$ in component form.

23) $$\quad \vecs a=⟨−1,−2,4⟩,\quad \vecs b=⟨−5,6,−7⟩$$

$$\vecs a+\vecs b=⟨−6,4,−3⟩, 4\vecs a=⟨−4,−8,16⟩, −5\vecs a+3\vecs b=⟨−10,28,−41⟩$$

24) $$\quad \vecs a=⟨3,−2,4⟩,\quad \vecs b=⟨−5,6,−9⟩$$

25) $$\quad \vecs a=−\hat{\mathbf k},\quad \vecs b=−\hat{\mathbf i}$$

$$\vecs a+\vecs b=⟨−1,0,−1⟩, 4\vecs a=⟨0,0,−4⟩, −5\vecs a+3\vecs b=⟨−3,0,5⟩$$

26) $$\quad \vecs a=\hat{\mathbf i}+\hat{\mathbf j}+\hat{\mathbf k},\quad \vecs b=2\hat{\mathbf i}−3\hat{\mathbf j}+2\hat{\mathbf k}$$

For exercises 27-30, vectors $$\vecs u$$ and $$\vecs v$$ are given. Find the magnitudes of vectors $$\vecs u−\vecs v$$ and $$−2\vecs u$$.

27) $$\quad \vecs u=2\hat{\mathbf i}+3\hat{\mathbf j}+4\hat{\mathbf k}, \quad \vecs v=−\hat{\mathbf i}+5\hat{\mathbf j}−\hat{\mathbf k}$$

$$\|\vecs u−\vecs v\|=\sqrt{38}, \quad \|−2\vecs u\|=2\sqrt{29}$$

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

29) $$\quad \vecs u=⟨2\cos t,−2\sin t,3⟩, \quad \vecs v=⟨0,0,3⟩,\quad$$ where $$t$$ is a real number.

$$\|\vecs u−\vecs v\|=2, \quad \|−2\vecs u\|=2\sqrt{13}$$

30) $$\quad \vecs u=⟨0,1,\sinh t⟩, \quad \vecs v=⟨1,1,0⟩,\quad$$ where $$t$$ is a real number.

For exercises 31-36, find the unit vector in the direction of the given vector $$\vecs a$$ and express it using standard unit vectors.

31) $$\quad \vecs a=3\hat{\mathbf i}−4\hat{\mathbf j}$$

$$\frac{3}{5}\hat{\mathbf i}−\frac{4}{5}\hat{\mathbf j}$$

32) $$\quad \vecs a=⟨4,−3,6⟩$$

33) $$\quad \vecs a=\vecd{PQ}$$, where $$P(−2,3,1)$$ and $$Q(0,−4,4)$$

$$\frac{\sqrt{62}}{31}\hat{\mathbf i}−\frac{7\sqrt{62}}{62}\hat{\mathbf j}+\frac{3\sqrt{62}}{62}\hat{\mathbf k}$$

34) $$\quad \vecs a=\vecd{OP},$$ where $$P(−1,−1,1)$$

35) $$\quad \vecs a=\vecs u−\vecs v+\vecs w,$$ where $$\vecs u=\hat{\mathbf i}−\hat{\mathbf j}−\hat{\mathbf k},\quad \vecs v=2\hat{\mathbf i}−\hat{\mathbf j}+\hat{\mathbf k}, \quad$$ and $$\vecs w=−\hat{\mathbf i}+\hat{\mathbf j}+3\hat{\mathbf k}$$

$$−\frac{\sqrt{6}}{3}\hat{\mathbf i}+\frac{\sqrt{6}}{6}\hat{\mathbf j}+\frac{\sqrt{6}}{6}\hat{\mathbf k}$$

36) $$\quad \vecs a=2\vecs u+\vecs v−\vecs w,\quad$$ where $$\vecs u=\hat{\mathbf i}−\hat{\mathbf k}, \quad \vecs v=2\hat{\mathbf j} \quad$$, and $$\vecs w=\hat{\mathbf i}−\hat{\mathbf j}$$

37) Determine whether $$\vecd{AB}$$ and $$\vecd{PQ}$$ are equivalent vectors, where $$A(1,1,1),\,B(3,3,3),\,P(1,4,5),$$ and $$Q(3,6,7).$$

Equivalent vectors

38) Determine whether the vectors $$\vecd{AB}$$ and $$\vecd{PQ}$$ are equivalent, where $$A(1,4,1),\, B(−2,2,0),\, P(2,5,7),$$ and $$Q(−3,2,1)$$.

For exercises 39-42, find vector $$\vecs u$$ with a magnitude that is given and satisfies the given conditions.

39) $$\quad \vecs v=⟨7,−1,3⟩, \, ‖\vecs u‖=10$$, and $$\vecs u$$ and $$\vecs v$$ have the same direction

$$\vecs u=⟨\frac{70\sqrt{59}}{59},−\frac{10\sqrt{59}}{59},\frac{30\sqrt{59}}{59}⟩$$

40) $$\quad \vecs v=⟨2,4,1⟩,\, ‖\vecs u‖=15$$, and $$\vecs u$$ and $$\vecs v$$ have the same direction

41) $$\quad \vecs v=⟨2\sin t,\, 2\cos t,1⟩, ‖\vecs u‖=2,\vecs u$$ and $$\vecs v$$ have opposite directions for any $$t$$, where $$t$$ is a real number

$$\vecs u=⟨−\frac{4\sqrt{5}}{5}\sin t,−\frac{4\sqrt{5}}{5}\cos t,−\frac{2\sqrt{5}}{5}⟩$$

42) $$\quad \vecs v=⟨3\sinh t,0,3⟩,\, ‖\vecs u‖=5$$, and $$\vecs u$$ and $$\vecs v$$ have opposite directions for any $$t$$, where $$t$$ is a real number

43) Determine a vector of magnitude $$5$$ in the direction of vector $$\vecd{AB}$$, where $$A(2,1,5)$$ and $$B(3,4,−7).$$

$$⟨\frac{5\sqrt{154}}{154},\frac{15\sqrt{154}}{154},−\frac{30\sqrt{154}}{77}⟩$$

44) Find a vector of magnitude $$2$$ that points in the opposite direction than vector $$\vecd{AB}$$, where $$A(−1,−1,1)$$ and $$B(0,1,1).$$ Express the answer in component form.

45) Consider the points $$A(2,α,0), \, B(0,1,β),$$ and $$C(1,1,β)$$, where $$α$$ and $$β$$ are negative real numbers. Find $$α$$ and $$β$$ such that $$\|\vecd{OA}−\vecd{OB}+\vecd{OC}\|=\|\vecd{OB}\|=4.$$

$$α=−\sqrt{7}, \,β=−\sqrt{15}$$

46) Consider points $$A(α,0,0),\,B(0,β,0),$$ and $$C(α,β,β),$$ where $$α$$ and $$β$$ are positive real numbers. Find $$α$$ and $$β$$ such that $$\|\overline{OA}+\overline{OB}\|=\sqrt{2}$$ and $$\|\overline{OC}\|=\sqrt{3}$$.

47) Let $$P(x,y,z)$$ be a point situated at an equal distance from points $$A(1,−1,0)$$ and $$B(−1,2,1)$$. Show that point $$P$$ lies on the plane of equation $$−2x+3y+z=2.$$

48) Let $$P(x,y,z)$$ be a point situated at an equal distance from the origin and point $$A(4,1,2)$$. Show that the coordinates of point P satisfy the equation $$8x+2y+4z=21.$$

49) The points $$A,B,$$ and $$C$$ are collinear (in this order) if the relation $${\|\vecd{AB}\|+\|\vecd{BC}\|=\|\vecd{AC}\|}$$ is satisfied. Show that $$A(5,3,−1),\, B(−5,−3,1),$$ and $$C(−15,−9,3)$$ are collinear points.

50) Show that points $$A(1,0,1), \, B(0,1,1),$$ and $$C(1,1,1)$$ are not collinear.

51) [T] A force $$\vecs F$$ of $$50 \,N$$ acts on a particle in the direction of the vector $$\vecd{OP}$$, where $$P(3,4,0).$$

a. Express the force as a vector in component form.

b. Find the angle between force $$\vecs F$$ and the positive direction of the $$x$$-axis. Express the answer in degrees rounded to the nearest integer.

$$a. \vecs F=⟨30,40,0⟩; \quad b. 53°$$

52) [T] A force $$\vecs F$$ of $$40\,N$$ acts on a box in the direction of the vector $$\vecd{OP}$$, where $$P(1,0,2).$$

a. Express the force as a vector by using standard unit vectors.

b. Find the angle between force $$\vecs F$$ and the positive direction of the $$x$$-axis.

53) If $$\vecs F$$ is a force that moves an object from point $$P_1(x_1,y_1,z_1)$$ to another point $$P_2(x_2,y_2,z_2)$$, then the displacement vector is defined as $$\vecs D=(x_2−x_1)\hat{\mathbf i}+(y_2−y_1)\hat{\mathbf j}+(z_2−z_1)\hat{\mathbf k}$$. A metal container is lifted $$10$$ m vertically by a constant force $$\vecs F$$. Express the displacement vector $$\vecs D$$ by using standard unit vectors.

$$\vecs D=10\hat{\mathbf k}$$

54) A box is pulled $$4$$ yd horizontally in the $$x$$-direction by a constant force $$\vecs F$$. Find the displacement vector in component form.

55) The sum of the forces acting on an object is called the resultant or net force. An object is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let $$\vecs F_1=⟨10,6,3⟩, \vecs F_2=⟨0,4,9⟩$$, and $$\vecs F_3=⟨10,−3,−9⟩$$ be three forces acting on a box. Find the force $$\vecs F_4$$ acting on the box such that the box is in static equilibrium. Express the answer in component form.

$$\vecs F_4=⟨−20,−7,−3⟩$$

56) [T] Let $$\vecs F_k=⟨1,k,k^2⟩, k=1,...,n$$ be $$n$$ forces acting on a particle, with $$n≥2.$$

a. Find the net force $$\vecs F=\sum_{k=1}^n\vecs F_k.$$ Express the answer using standard unit vectors.

b. Use a computer algebra system (CAS) to find $$n$$ such that $$\|\vecs F\|<100.$$

57) The force of gravity $$\vecs F$$ acting on an object is given by $$\vecs F=m\vecs g$$, where $$m$$ is the mass of the object (expressed in kilograms) and $$\vecs g$$ is acceleration resulting from gravity, with $$\|\vecs g\|=9.8 \,N/kg.$$ A 2-kg disco ball hangs by a chain from the ceiling of a room.

a. Find the force of gravity $$\vecs F$$ acting on the disco ball and find its magnitude.

b. Find the force of tension $$\vecs T$$ in the chain and its magnitude.

Express the answers using standard unit vectors.

$$a. \vecs F=−19.6\hat{\mathbf k}, \quad \|\vecs F\|=19.6 \,N$$
$$b. \vecs T=19.6\hat{\mathbf k}, \quad \|\vecs T\|=19.6 \,N$$

58) A 5-kg pendant chandelier is designed such that the alabaster bowl is held by four chains of equal length, as shown in the following figure.

a. Find the magnitude of the force of gravity acting on the chandelier.

b. Find the magnitudes of the forces of tension for each of the four chains (assume chains are essentially vertical).

59) [T] A 30-kg block of cement is suspended by three cables of equal length that are anchored at points $$P(−2,0,0), Q(1,\sqrt{3},0),$$ and $$R(1,−\sqrt{3},0)$$. The load is located at $$S(0,0,−2\sqrt{3})$$, as shown in the following figure. Let $$\vecs F_1, \vecs F_2$$, and $$\vecs F_3$$ be the forces of tension resulting from the load in cables $$RS,QS,$$ and $$PS,$$ respectively.

a. Find the gravitational force $$\vecs F$$ acting on the block of cement that counterbalances the sum $$\vecs F_1+\vecs F_2+\vecs F_3$$ of the forces of tension in the cables.

b. Find forces $$\vecs F_1, \vecs F_2,$$ and $$\vecs F_3$$. Express the answer in component form.

a. $$\vecs F=−294\hat{\mathbf k}$$ N;
b. $$\vecs F_1=⟨−\frac{49\sqrt{3}}{3},49,−98⟩, \vecs F_2=⟨−\frac{49\sqrt{3}}{3},−49,−98⟩$$, and $$\vecs F_3=⟨\frac{98\sqrt{3}}{3},0,−98⟩$$ (each component is expressed in newtons)

60) Two soccer players are practicing for an upcoming game. One of them runs 10 m from point A to point B. She then turns left at $$90°$$ and runs 10 m until she reaches point C. Then she kicks the ball with a speed of 10 m/sec at an upward angle of $$45°$$ to her teammate, who is located at point A. Write the velocity of the ball in component form.

61) Let $$\vecs r(t)=⟨x(t),\, y(t), \, z(t)⟩$$ be the position vector of a particle at the time $$t∈[0,T]$$, where $$x,y,$$ and $$z$$ are smooth functions on $$[0,T]$$. The instantaneous velocity of the particle at time $$t$$ is defined by vector $$\vecs v(t)=⟨x'(t), \, y'(t), \, z'(t)⟩$$, with components that are the derivatives with respect to $$t$$, of the functions $$x, y$$, and $$z$$, respectively. The magnitude $$∥\vecs v(t)∥$$ of the instantaneous velocity vector is called the speed of the particle at time $$t$$. Vector $$\vecs a(t)=⟨x''(t), \, y''(t), \, z''(t)⟩$$, with components that are the second derivatives with respect to $$t$$, of the functions $$x,y,$$ and $$z$$, respectively, gives the acceleration of the particle at time $$t$$. 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 is expressed in seconds.

a. Find the instantaneous velocity, speed, and acceleration of the particle after the first second. Round your answer to two decimal places.

b. Use a CAS to visualize the path of the particle—that is, the set of all points of coordinates $$(\cos t,\sin t,2t),$$ where $$t∈[0,30].$$

$$a. \vecs v(1)=⟨−0.84,0.54,2⟩$$ (each component is expressed in centimeters per second); $$∥\vecs v(1)∥=2.24$$ (expressed in centimeters per second); $$\vecs a(1)=⟨−0.54,−0.84,0⟩$$ (each component expressed in centimeters per second squared);

$$b.$$

62) [T] Let $$\vecs r(t)=⟨t,2t^2,4t^2⟩$$ be the position vector of a particle at time $$t$$ (in seconds), where $$t∈[0,10]$$ (here the components of $$\vecs r$$ are expressed in centimeters).

a. Find the instantaneous velocity, speed, and acceleration of the particle after the first two seconds. Round your answer to two decimal places.

b. Use a CAS to visualize the path of the particle defined by the points $$(t, \, 2t^2, \, 4t^2),$$ where $$t∈[0, \, 60].$$