crest Vectors
 Quiz 9

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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 (to one decimal place)?



Your answer is correct

Since u · v = |u||v| cos h, where h is the angle between the vectors when placed tail to tail, we have u · v = 2 × 3 × cos 45 ~~ 4.24 .

Not correct. You may try again.

Use the fact that u · v = |u||v| cos h. The answer is 4.2.

Question 2

What is the angle (to two decimal places) between a and b if a · b = 3, |a| = 2, and |b| = 2.6?

radians

Your answer is correct

If h is the required angle, then cos h = a·b |a||b| = 3 5.2- and hence h ~~ 0.96 radians.

Not correct. You may try again.

If h is the required angle, then cos h = a·b- |a||b| = 3-- 5.2

Question 3

What is a·b if a = 3i-j and b = 2i + j + 4k?



Your answer is correct

a·b = 3 × 2 - 1 × 1 + 0 × 4 = 5.

Not correct. You may try again.

The answer is 5

Question 4

Express u in Cartesian form as ai + bj given that u · u = 12 and u points towards the north-west.
1. 12i + 12j
2. 6i - 6j
3. - V~ --- 12i + V~ --- 12j
4. - V~ -- 6i + V~ - 6j

Not correct. Choice 1 is false.

Not correct. Choice 2 is false.

Not correct. Choice 3 is false.

Your answer is correct

Since u · u = |u|2 = 12, we have |u| = 2 V~ -- 3. Hence
V~ -- V~ -- 1 V~ -- V~ -- u = 2 3 ^u = 2 3( V~ -(- i + j) = - 6i + 6j. 2

Question 5

If u = 5j and u · v = 0, what conclusion can be drawn?
1. v points east or west.
2. v points south.
3. v is parallel to u.
4. None of the above is strictly true.

Not correct. Choice 1 is false.

Not correct. Choice 2 is false.

Not correct. Choice 3 is false.

Your answer is correct

When the scalar product of two vectors is zero, we can conclude that either one or both of the vectors is the zero vector, or else that both the vectors are non-zero and they are mutually perpendicular. In this case, we are told that u = 5j and so u is non-zero. Hence either v is the zero vector or v is a non-zero vector perpendicular to u (that is, v points east or west). If the first statement were amended to include the possibility that v is the zero vector then it would be the best answer to the question.

Question 6

What is the following expression?
(u + 3v × a) · u
1. a vector
2. a scalar
3. not defined

Not correct. Choice 1 is false.

The expression inside the brackets must be a vector in order for the scalar product to be meaningful, and indeed it is. However, the resulting entire expression is not a vector.

Not correct. Choice 2 is false.

The expression inside the brackets must be a vector in order for the scalar product to be meaningful, and indeed it is. The resulting expression is therefore a scalar.

Not correct. Choice 3 is false.

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