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Quiz 2: Unit vectors

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Question 1

Let i, j and k be unit vectors in the x, y and z directions, respectively. Suppose that a = 3i - j + 2k. What is the magnitude of the vector a?

a)
4
   b)
√ -- 12
c)
√ -- 14
   d)
14

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
|a| = ∘32-+-(- 1)2 +-22 = √9-+-1-+-4 = √14-.
Not correct. Choice (d) is false.

Question 2

Suppose that a = 2i + j, b = i- 4j + k, and c = j + k. What is the magnitude of the vector 2a - b + c  ?

a)
√-- 58
   b)
10
c)
2√ - 5 -√ -- 18 + √- 2
   d)
√ -- 30

 

Your answer is correct.
We have 2a - b + c = 3i + 7j, and so |2a - b + c| = √32 +-72 = √58.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 3

Which of the of 3-dimension coordinates axes below are right–handed coordinate systems? (The arrows indicate the positive direction on each axis.)

a)
PIC
   b)
PIC
c)
PIC
   d)
PIC
e)
PIC

 

There is at least one mistake.
For example, choice (a) should be true.
Using the right hand rule, you should have your thumb pointing up in the direction of the z–axis, your index finger pointing in the direction of the x–axis and your middle finger in the direction of the y–axis.
There is at least one mistake.
For example, choice (b) should be false.
When you put the z–axis on your thumb, the y–axis on your middle finger and the x–axis on your index finger your thumb is pointing down. Therefore, this coordinate system is left handed.
There is at least one mistake.
For example, choice (c) should be false.
When you put the z–axis on your thumb, the y–axis on your middle finger and the x–axis on your index finger your thumb is pointing down. Therefore, this coordinate system is left handed.
There is at least one mistake.
For example, choice (d) should be true.
When you put the z–axis on your thumb, the y–axis on your middle finger and the x–axis on your index finger your thumb is pointing up. Therefore, this coordinate system is right handed.
There is at least one mistake.
For example, choice (e) should be true.
This is the standard 3D coordinate system, so it is right handed.
Your answers are correct
  1. True. Using the right hand rule, you should have your thumb pointing up in the direction of the z–axis, your index finger pointing in the direction of the x–axis and your middle finger in the direction of the y–axis.
  2. False. When you put the z–axis on your thumb, the y–axis on your middle finger and the x–axis on your index finger your thumb is pointing down. Therefore, this coordinate system is left handed.
  3. False. When you put the z–axis on your thumb, the y–axis on your middle finger and the x–axis on your index finger your thumb is pointing down. Therefore, this coordinate system is left handed.
  4. True. When you put the z–axis on your thumb, the y–axis on your middle finger and the x–axis on your index finger your thumb is pointing up. Therefore, this coordinate system is right handed.
  5. True. This is the standard 3D coordinate system, so it is right handed.

Question 4

Given that √ - u = 3i- 2j+ 3k , find the unit vector in the opposite direction to u.

a)
√ - --3i- 1j+ 3 k 4 2 4
  b)
i+ j+ k
c)
√3- 1 3 - -4-i+ 2j- 4k
  d)
√3- 2 3 √2-i- √2-j+ √2-k

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
First, find the unit vector in the same direction as u. This requires knowing the magnitude of u, which is |u| = √3-+-4-+9- = 4. Then 1 4u is the unit vector in the same direction as u, so its negative, -1 - 4u, is the answer.
Not correct. Choice (d) is false.

Question 5

Given that u = 3i + 2j - k, v = i - 3j + 2k, find the unit vector in the direction of u - 2v.

a)
-√1--(i+8j - 5k ) 3 10
  b)
1 -√---(i+ 8j- 5k) 3 14
c)
--1- √14-(i+ 8j- 5k)
  d)
√-1-(2i+ 5j+ k) 14

 

Your answer is correct.
u - 2v = i + 8j - 5k.
|u - 2v| = √---------- 1+ 64 + 25 = √ -- 90 = 3√ -- 10
The unit vector in the direction of u - 2v is therefore
-u--2v- --1-- |u- 2v| = 3√10(i+ 8j- 5k).
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 6

Find the polar angle (in radians) of - 2i- j .

a)
0.463
   b)
3.605
c)
2.677
   d)
-0.463

 

Not correct. Choice (a) is false.
Your answer is correct.
If we set -O-→P = -2i - j, then the point P is in the third quadrant. Hence if θ is a polar angle of --→ OP, then tanθ = 1 2, and so θ = π + tan-11 2 ≈ 3.605 radians.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 7

Find the polar angle (in radians) of -√ - 3i + j.

a)
π -- 6
   b)
- π- 6
c)
5π --- 6
   d)
- 5π- 6

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
1-- tanθ = - √3 in the second quadrant so θ = 5π- 6.
Not correct. Choice (d) is false.

Question 8

Relative to the origin, point P has position vector u and Q has position vector v. What is --Q→P ?

a)
u - v
   b)
v - u
c)
u + v
   d)
-u - v

 

Your answer is correct.
-Q-→P = --O→P - -O-→Q = u- v
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 9

Relative to the origin, point P has position vector 3i + j - 4k and Q has position vector i - j + 3k. What is -Q-P→   ?

a)
2i + 2j - 7k
   b)
2i - 7k
c)
3i - j - 12k
   d)
-2i - 2j + 7k

 

Your answer is correct.
Since -Q-→P = --O→P - -O-→Q , we have -Q-→P = 3i + j - 4k - (i - j + 3k) = 2i + 2j - 7k.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 10

A boat sails 5 km south-east then 3 km due west. Approximately how far from its starting position is it now ?

a)
8 km
   b)
4.2 km
c)
7.4 km
   d)
3.6 km

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
The boat’s journey can be represented by the following diagram, where the origin O is taken to be the starting position and Q is the finishing position.
PQ0
The required distance from the starting position is then |-O-→Q|. Now as --→ OP = 5-- √2i -5-- √2j, and --→ PQ = -3i, we have
--→ --→ --→ (-5- ) -5- OQ = OP + P Q = √2 - 3 i- √2-j.
The magnitude of this vector is the required distance, namely
∘ ---------------- 25+ 9- 3√0-+ 25 ≈ 3.6 km. 2 2 2
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