# Quiz 3: Dot products of vectors

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 $4{5}^{\circ }$, what is $u\cdot v$ ?
 a) 4.5 b) 6.2 c) 4.2 d) 5.1

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Since $u\cdot v=|u||v|cos\theta$, where $\theta$ is the angle between the vectors when placed tail to tail, we have $u\cdot v=2×3×cos4{5}^{\circ }\approx 4.24.$
Not correct. Choice (d) is false.

## Question 2

What is the approximate angle between $a$ and $b$ if $a\cdot b=3$, $|a|=2$, $|b|=2.6$ ?

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
If $\theta$ is the required angle, then $cos\theta =\frac{a\cdot b}{|a||b|}=\frac{3}{5.2}$ and hence $\theta \approx 0.955$ radians.

## Question 3

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

Not correct. Choice (a) is false.
$a\cdot b=3×2-1×1+0×4=5.$
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

## Question 4

Suppose that $u$ is a vector pointing north-west with $u\cdot u=12$. 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) $-\sqrt{12}i+\sqrt{12}j$ d) $-\sqrt{6}i+\sqrt{6}j$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Since $u\cdot u=|u{|}^{2}=12$, we have $|u|=2\sqrt{3}.$ Hence
$u=2\sqrt{3}\phantom{\rule{0.3em}{0ex}}\stackrel{^}{u}=2\sqrt{3}\phantom{\rule{0.3em}{0ex}}\left(\frac{-i+j}{\sqrt{2}}\right)=-\sqrt{6}i+\sqrt{6}j.$

## 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 $u\cdot v=0$. 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 $u\cdot v=0$ 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 $u\cdot v$ 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 $u\cdot v\ne 0$.
There is at least one mistake.
For example, choice (d) should be true.
Since $u\cdot v=0$ and $v$ is a non-zero vector $u$ and $v$ must be perpendicular to each other.
1. True. Since $u\cdot v=0$ we know that $u$ and $v$ are perpendicular.
2. False. Since $u$ points north, if $v$ pointed south then $u\cdot v$ would be negative and, in particular, non-zero.
3. False. If $u$ and $v$ were parallel then $u\cdot v\ne 0$.
4. True. Since $u\cdot v=0$ 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 $\theta$ 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 $a\cdot \stackrel{^}{b}=|a|cos\theta$. Therefore, $cos\theta$ is negative and $\theta$ is obtuse angle. The picture looks something like the following, where the blue vector is the component of $a$ in the direction of $b$.
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.
1. False. Try drawing a diagram.
2. True. Let $\theta$ 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 $a\cdot \stackrel{^}{b}=|a|cos\theta$. Therefore, $cos\theta$ is negative and $\theta$ is obtuse angle. The picture looks something like the following, where the blue vector is the component of $a$ in the direction of $b$.
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) $-0.51$

The required component is the number $a\cdot \stackrel{^}{b}$. Now since $|b|=\sqrt{41}$, this gives
$a\cdot \stackrel{^}{b}=\left(3i+j-k\right)\cdot \frac{1}{\sqrt{41}}\left(i-2j+6k\right)=\frac{3-2-6}{\sqrt{41}}\approx -0.78.$
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) $-\frac{1}{25}\left(4j-3k\right)$ b) $\frac{7}{25}\left(4j-3k\right)$ c) $\frac{1}{25}\left(4j-3k\right)$ d) $-\frac{3}{25}\left(4j-3k\right)$

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
The required projection is $\left(u\cdot \stackrel{^}{a}\right)\stackrel{^}{a}$. Since $\stackrel{^}{a}=\frac{1}{5}\left(4j-3k\right)$, we have $u\cdot \stackrel{^}{a}=\frac{1}{5}\left(4-3\right)=\frac{1}{5},$ and so $\left(u\cdot \stackrel{^}{a}\right)\stackrel{^}{a}=\frac{1}{25}\left(4j-3k\right)$.
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 $\frac{\pi }{4}$ radians, or $45$ degrees. In which direction is the vector $u×v$ pointing?
 (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.
The direction of $u×v$ in (1) is different from that in (2).
Not correct. Choice (b) is false.
The direction of $u×v$ in (1) is different from that in (2).
Not correct. Choice (d) is false.
The direction of $u×v$ is such that $u,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}v,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}u×v$ 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) $4j+2k$

Using the component form of the vector cross product formula, if $a={a}_{1}i+{a}_{2}j+{a}_{3}k$ and $b={b}_{1}i+{b}_{2}j+{b}_{3}k$, then $a×b=\left({a}_{2}{b}_{3}-{a}_{3}{b}_{2}\right)i-\left({a}_{1}{b}_{3}-{a}_{3}{b}_{1}\right)j+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}\right)k.$ This gives $a×b=-2i-8j-7k.$