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Quiz 3: Dot products of vectors

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Given that u is a vector of magnitude 2, v is a vector of magnitude 3 and the angle between them when placed tail to tail is 45∘ , what is u⋅v ?
a) 4.5   b) 6.2
c) 4.2   d) 5.1

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Since u⋅v = ∣u∣∣v∣cosθ, where θ is the angle between the vectors when placed tail to tail, we have u ⋅ v = 2 × 3 × cos45∘≈ 4.24.
Not correct. Choice (d) is false.
 

Question 2

 
 
What is the approximate angle between a and b if a⋅b = 3 , ∣a∣ = 2 , ∣b∣ = 2.6 ?
a) 1.25 radians   b) 0.87 radians
c) 1.32 radians   d) 0.96 radians

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
If θ is the required angle, then cosθ = a∣a⋅∣b∣b∣ = 53.2 and hence θ ≈ 0.955 radians.
 

Question 3

 
 
What is a⋅b if a = 3i- j and b = 2i + j + 4k  ?
a) 3   b) 5
c) -2   d) 0.4

 

Not correct. Choice (a) is false.
Your answer is correct.
a ⋅ b = 3 × 2 - 1 × 1 + 0 × 4 = 5.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 4

 
 
Suppose that u is a vector pointing north-west with u ⋅u = 12 . Which of the following vectors is equal to u written in Cartesian form? (Here the unit vector i points towards the east and the unit vector j points north.)
a) 12i+ 12j   b) 6i- 6j
c) √-- √ -- - 12i+ 12j   d) √- √ - - 6i+ 6j

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Since 2 u ⋅u = ∣u∣ = 12 , we have √ - ∣u∣ = 2 3. Hence
√-ˆ √ - -i√+j √ - √- u = 2 3u = 2 3( 2 ) = - 6i+ 6j.
 

Question 5

 
 
Assume that the unit vector i points towards the east and the unit vector j points north. Suppose that we are given two non-zero vectors u and v such that u = 5j and u ⋅v = 0 . Which of the following statements must be true? (More than one answer may be correct.)
a) v points east or west.   b) v points south.
c) v is parallel to u.   d) v is perpendicular to u.

 

There is at least one mistake.
For example, choice (a) should be true.
Since u ⋅ v = 0 we know that u and v are perpendicular.
There is at least one mistake.
For example, choice (b) should be false.
Since u points north, if v pointed south then u⋅v would be negative and, in particular, non-zero.
There is at least one mistake.
For example, choice (c) should be false.
If u and v were parallel then u ⋅ v⁄=0.
There is at least one mistake.
For example, choice (d) should be true.
Since u ⋅ v = 0 and v is a non-zero vector u and v must be perpendicular to each other.
Your answers are correct
  1. True. Since u ⋅ v = 0 we know that u and v are perpendicular.
  2. False. Since u points north, if v pointed south then u⋅v would be negative and, in particular, non-zero.
  3. False. If u and v were parallel then u ⋅ v⁄=0.
  4. True. Since u ⋅ v = 0 and v is a non-zero vector u and v must be perpendicular to each other.
 

Question 6

 
 
Let a and b be two vectors. If the component of a in the direction of b is negative this means:
a) There is an arithmetical error in the calculation.
b) The angle between a and b is obtuse, when the vectors are placed tail to tail.
c) The angle between a and b when placed head to tail is acute.
d) None of the above.

 

There is at least one mistake.
For example, choice (a) should be false.
Try drawing a diagram.
There is at least one mistake.
For example, choice (b) should be true.
Let θ be the angle between a and b when the two vectors are placed tail to tail. Then the component of a in the direction of b is given by a⋅bˆ= ∣a∣cosθ . Therefore, cosθ is negative and θ is obtuse angle. The picture looks something like the following, where the blue vector is the component of a in the direction of b.
ba
There is at least one mistake.
For example, choice (c) should be true.
This is equivalent to response (b).
There is at least one mistake.
For example, choice (d) should be false.
Try drawing a diagram.
Your answers are correct
  1. False. Try drawing a diagram.
  2. True. Let θ be the angle between a and b when the two vectors are placed tail to tail. Then the component of a in the direction of b is given by a⋅bˆ= ∣a∣cosθ . Therefore, cosθ is negative and θ is obtuse angle. The picture looks something like the following, where the blue vector is the component of a in the direction of b.
    ba
  3. True. This is equivalent to response (b).
  4. False. Try drawing a diagram.
 

Question 7

 
 
What is the component of a = 3i + j - k in the direction of b = i - 2j + 6k  ?
a) -0.78   b) 1.23
c) -0.34   d) -0.51

 

Your answer is correct.
The required component is the number a ⋅ˆ b. Now since ∣b∣ = √-- 41, this gives
ˆ √1- 3-√2-6 a ⋅b = (3i+ j- k) ⋅ 41(i- 2j+ 6k) = 41 ≈ - 0.78.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 8

 
 
If u = 3i + j + k and a = 4j - 3k, find the projection of u in the direction of a.
a) - 125(4j- 3k)   b) 275(4j- 3k)
c) 215(4j- 3k)   d) -3 25(4j - 3k)

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
The required projection is (u ⋅ˆa)ˆa . Since 1 ˆa = 5(4j- 3k) , we have
u ⋅ˆa = 1(4- 3) = 1, 5 5
and so (u ⋅ˆa)ˆa = 125(4j- 3k) .
Not correct. Choice (d) is false.
 

Question 9

 
 
In both of the diagrams below, the vectors u and v lie in the xy plane in 3-dimensional space and the angle between them is π- 4 radians, or 45 degrees. In which direction is the vector u× v pointing?
xyzuv xyzvu (1) (2)
a) In the positive z direction in both (1) and (2).
b) In the negative z direction in both (1) and (2).
c) In the positive z direction in (1) and the negative z direction in (2).
d) In the positive z direction in (2) and the negative z direction in (1).

 

Not correct. Choice (a) is false.
As u×v = -v ×u these two vectors point in dirrect directions.
Not correct. Choice (b) is false.
As u×v = -v ×u these two vectors point in dirrect directions.
Your answer is correct.
Not correct. Choice (d) is false.
The direction of u × v is such that u, v, u × v form a right-hand set.
 

Question 10

 
 
Calculate the vector cross product a × b when a = 3i + j - 2k and b = 4i - j.
a) -2i - 8j - 7k   b) -2i + 8j - 7k
c) 2i - 8j + 7k   d) 4j + 2k

 

Your answer is correct.
Using the component form of the vector cross product formula, if a = a1i + a2j + a3k and b = b1i + b2j + b3k, then
a× b = (a2b3 - a3b2)i- (a1b3 - a3b1)j+ (a1b2 - a2b1)k.
This gives a × b = -2i - 8j - 7k.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.