7.4e: Vectors in 3D

A:  3-D Vector Practice

Exercise \(\PageIndex{A}\):  Vector Overview 

1) Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point \(\displaystyle A(2,3,5)\) 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 \(\displaystyle O\) and \(\displaystyle A\).

 

 

2) Find the coordinates of point \(\displaystyle P\) and determine its distance to the origin.

 

 

 

Express vector \( \vecd{PQ} \) with the initial point at \(\displaystyle P\) and the terminal point at \(\displaystyle Q\)
(a). in component form and (b). by using standard unit vectors.

3) \(\displaystyle P(3,0,2)\) and \(\displaystyle Q(−1,−1,4)\)

4) \(\displaystyle P(0,10,5)\) and \(\displaystyle Q(1,1,−3)\)

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

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

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

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

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.

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

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

13) \(\quad \vecs a=−\hat{\mathbf k},\quad \vecs b=−\hat{\mathbf i}\)

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

Given vectors \(\vecs u\) and \(\vecs v\) below, find the magnitudes of vectors \(\vecs u−\vecs v\) and \(−2\vecs u\).

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

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

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

24) \(\quad \vecs u=⟨0,1,\sin  t⟩, \quad \vecs v=⟨1,1,0⟩,\quad\) where \(t\) is a real number.

Find the unit vector in the direction of the given vector \(\displaystyle a\) and express it using standard unit vectors.

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

32) \(\quad \vecs a=⟨4,−3,6⟩\)

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

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 \(\vecs w=−\hat{\mathbf i}+\hat{\mathbf j}+3\hat{\mathbf k}\)

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 \(Q(3,6,7).\)

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

Find vector \( \vecs u\) with a magnitude that is given and which satisfies the given conditions.

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

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

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

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

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

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

Answers to odd exercises: 

1. \(\displaystyle a. (2,0,5),(2,0,0),(2,3,0),(0,3,0),(0,3,5),(0,0,5); b. \sqrt{38}\)
3. \(a. \vecd{PQ}=⟨−4,−1,2⟩\) \( \quad \) \(b. \vecd{PQ}=−4\hat{\mathbf i}−\hat{\mathbf j}+2\hat{\mathbf k}\)
5. \(a. \vecd{PQ}=⟨6,−24,24⟩\) \( \quad \) \(b. \vecd{PQ}=6\hat{\mathbf i}−24\hat{\mathbf j}+24\hat{\mathbf k}\)
7. \(\displaystyle Q(5,2,8)\)
11. \(\vecs a+\vecs b=⟨−6,4,−3⟩, 4\vecs a=⟨−4,−8,16⟩, −5\vecs a+3\vecs b=⟨−10,28,−41⟩\)
13. \(\vecs a+\vecs b=⟨−1,0,−1⟩, 4\vecs a=⟨0,0,−4⟩, −5\vecs a+3\vecs b=⟨−3,0,5⟩\)
21. \(\|\vecs u−\vecs v\|=\sqrt{38}, \quad \|−2\vecs u\|=2\sqrt{29}\)
23. \(\|\vecs u−\vecs v\|=2, \quad \|−2\vecs u\|=2\sqrt{13}\)
31. \(\frac{3}{5}\hat{\mathbf i}−\frac{4}{5}\hat{\mathbf j}\)
33. \(\frac{\sqrt{62}}{31}\hat{\mathbf i}−\frac{7\sqrt{62}}{62}\hat{\mathbf j}+\frac{3\sqrt{62}}{62}\hat{\mathbf k}\)
35. \(−\frac{\sqrt{6}}{3}\hat{\mathbf i}+\frac{\sqrt{6}}{6}\hat{\mathbf j}+\frac{\sqrt{6}}{6}\hat{\mathbf k}\)
37. Equivalent vectors
41. \(\vecs u=⟨\frac{70\sqrt{59}}{59},−\frac{10\sqrt{59}}{59},\frac{30\sqrt{59}}{59}⟩\)
43. \(\vecs u=⟨−\frac{4\sqrt{5}}{5}\sin t,−\frac{4\sqrt{5}}{5}\cos t,−\frac{2\sqrt{5}}{5}⟩\)
45. \(⟨\frac{5\sqrt{154}}{154},\frac{15\sqrt{154}}{154},−\frac{30\sqrt{154}}{77}⟩\)
47. \(α=−\sqrt{7}, \,β=−\sqrt{15}\)