MATH1002 Quizzes

Quiz 3: Dot products of vectors
Question 1 Questions
Given that $\mathbf{u}$ is a vector of magnitude 2, $\mathbf{v}$ is a vector of magnitude 3 and the angle between them when placed tail to tail is $4{5}^{\circ }$, what is $\mathbf{u}\cdot \mathbf{v}$ ? Exactly one option must be correct)
 a) 4.5 b) 6.2 c) 4.2 d) 5.1

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Since $\mathbf{u}\cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}|cos\theta$, where $\theta$ is the angle between the vectors when placed tail to tail, we have $\mathbf{u}\cdot \mathbf{v}=2×3×cos4{5}^{\circ }\approx 4.24.$
Choice (d) is incorrect
What is the approximate angle between $\mathbf{a}$ and $\mathbf{b}$ if $\mathbf{a}\cdot \mathbf{b}=3$, $|\mathbf{a}|=2$, $|\mathbf{b}|=2.6$ ? Exactly one option must be correct)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
If $\theta$ is the required angle, then $cos\theta =\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}=\frac{3}{5.2}$ and hence $\theta \approx 0.955$ radians.
What is $\mathbf{a}\cdot \mathbf{b}$ if $\mathbf{a}=3\mathbf{i}-\mathbf{j}$ and $\mathbf{b}=2\mathbf{i}+\mathbf{j}+4\mathbf{k}$ ? Exactly one option must be correct)
 a) 3 b) 5 c) $-2$ d) 0.4

Choice (a) is incorrect
Choice (b) is correct!
$\mathbf{a}\cdot \mathbf{b}=3×2-1×1+0×4=5.$
Choice (c) is incorrect
Choice (d) is incorrect
Suppose that $\mathbf{u}$ is a vector pointing north-west with $\mathbf{u}\cdot \mathbf{u}=12$. Which of the following vectors is equal to $\mathbf{u}$ written in Cartesian form? (Here the unit vector $\mathbf{i}$ points towards the east and the unit vector $\mathbf{j}$ points north.) Exactly one option must be correct)
 a) $12\mathbf{i}+12\mathbf{j}$ b) $6\mathbf{i}-6\mathbf{j}$ c) $-\sqrt{12}\mathbf{i}+\sqrt{12}\mathbf{j}$ d) $-\sqrt{6}\mathbf{i}+\sqrt{6}\mathbf{j}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Since $\mathbf{u}\cdot \mathbf{u}=|\mathbf{u}{|}^{2}=12$, we have $|\mathbf{u}|=2\sqrt{3}.$ Hence
$\mathbf{u}=2\sqrt{3}\phantom{\rule{0.3em}{0ex}}\stackrel{̂}{\mathbf{u}}=2\sqrt{3}\phantom{\rule{0.3em}{0ex}}\left(\frac{-\mathbf{i}+\mathbf{j}}{\sqrt{2}}\right)=-\sqrt{6}\mathbf{i}+\sqrt{6}\mathbf{j}.$
Assume that the unit vector $\mathbf{i}$ points towards the east and the unit vector $\mathbf{j}$ points north. Suppose that we are given two non-zero vectors $\mathbf{u}$ and $\mathbf{v}$ such that $\mathbf{u}=5\mathbf{j}$ and $\mathbf{u}\cdot \mathbf{v}=0$. Which of the following statements must be true? (More than one answer may be correct.) (Zero or more options can be correct)
 a) $\mathbf{v}$ points east or west. b) $\mathbf{v}$ points south. c) $\mathbf{v}$ is parallel to $\mathbf{u}$. d) $\mathbf{v}$ is perpendicular to $\mathbf{u}$.

There is at least one mistake.
For example, choice (a) should be True.
Since $\mathbf{u}\cdot \mathbf{v}=0$ we know that $\mathbf{u}$ and $\mathbf{v}$ are perpendicular.
There is at least one mistake.
For example, choice (b) should be False.
Since $\mathbf{u}$ points north, if $\mathbf{v}$ pointed south then $\mathbf{u}\cdot \mathbf{v}$ would be negative and, in particular, non-zero.
There is at least one mistake.
For example, choice (c) should be False.
If $\mathbf{u}$ and $\mathbf{v}$ were parallel then $\mathbf{u}\cdot \mathbf{v}\ne 0$.
There is at least one mistake.
For example, choice (d) should be True.
Since $\mathbf{u}\cdot \mathbf{v}=0$ and $\mathbf{v}$ is a non-zero vector $\mathbf{u}$ and $\mathbf{v}$ must be perpendicular to each other.
Correct!
1. True Since $\mathbf{u}\cdot \mathbf{v}=0$ we know that $\mathbf{u}$ and $\mathbf{v}$ are perpendicular.
2. False Since $\mathbf{u}$ points north, if $\mathbf{v}$ pointed south then $\mathbf{u}\cdot \mathbf{v}$ would be negative and, in particular, non-zero.
3. False If $\mathbf{u}$ and $\mathbf{v}$ were parallel then $\mathbf{u}\cdot \mathbf{v}\ne 0$.
4. True Since $\mathbf{u}\cdot \mathbf{v}=0$ and $\mathbf{v}$ is a non-zero vector $\mathbf{u}$ and $\mathbf{v}$ must be perpendicular to each other.
Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors. If the component of $\mathbf{a}$ in the direction of $\mathbf{b}$ is negative this means: (Zero or more options can be correct)
 a) There is an arithmetical error in the calculation. b) The angle between $\mathbf{a}$ and $\mathbf{b}$ is obtuse, when the vectors are placed tail to tail. c) The angle between $\mathbf{a}$ and $\mathbf{b}$ 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 $\mathbf{a}$ and $\mathbf{b}$ when the two vectors are placed tail to tail. Then the component of $\mathbf{a}$ in the direction of $\mathbf{b}$ is given by $\mathbf{a}\cdot \stackrel{̂}{\mathbf{b}}=|\mathbf{a}|cos\theta$. 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 $\mathbf{a}$ in the direction of $\mathbf{b}$.
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.
Correct!
1. False Try drawing a diagram.
2. True Let $\theta$ be the angle between $\mathbf{a}$ and $\mathbf{b}$ when the two vectors are placed tail to tail. Then the component of $\mathbf{a}$ in the direction of $\mathbf{b}$ is given by $\mathbf{a}\cdot \stackrel{̂}{\mathbf{b}}=|\mathbf{a}|cos\theta$. 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 $\mathbf{a}$ in the direction of $\mathbf{b}$.
3. True This is equivalent to response (b).
4. False Try drawing a diagram.
What is the component of $\mathbf{a}=3\mathbf{i}+\mathbf{j}-\mathbf{k}$ in the direction of $\mathbf{b}=\mathbf{i}-2\mathbf{j}+6\mathbf{k}$ ? Exactly one option must be correct)
 a) $-0.78$ b) 1.23 c) $-0.34$ d) $-0.51$

Choice (a) is correct!
The required component is the number $\mathbf{a}\cdot \stackrel{̂}{\mathbf{b}}$. Now since $|\mathbf{b}|=\sqrt{41}$, this gives
$\mathbf{a}\cdot \stackrel{̂}{\mathbf{b}}=\left(3\mathbf{i}+\mathbf{j}-\mathbf{k}\right)\cdot \frac{1}{\sqrt{41}}\left(\mathbf{i}-2\mathbf{j}+6\mathbf{k}\right)=\frac{3-2-6}{\sqrt{41}}\approx -0.78.$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
If $\mathbf{u}=3\mathbf{i}+\mathbf{j}+\mathbf{k}$ and $\mathbf{a}=4\mathbf{j}-3\mathbf{k}$, find the projection of $\mathbf{u}$ in the direction of $\mathbf{a}$. Exactly one option must be correct)
 a) $-\frac{1}{25}\left(4\mathbf{j}-3\mathbf{k}\right)$ b) $\frac{7}{25}\left(4\mathbf{j}-3\mathbf{k}\right)$ c) $\frac{1}{25}\left(4\mathbf{j}-3\mathbf{k}\right)$ d) $-\frac{3}{25}\left(4\mathbf{j}-3\mathbf{k}\right)$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The required projection is $\left(\mathbf{u}\cdot \stackrel{̂}{\mathbf{a}}\right)\stackrel{̂}{\mathbf{a}}$. Since $\stackrel{̂}{\mathbf{a}}=\frac{1}{5}\left(4\mathbf{j}-3\mathbf{k}\right)$, we have $\mathbf{u}\cdot \stackrel{̂}{\mathbf{a}}=\frac{1}{5}\left(4-3\right)=\frac{1}{5},$ and so $\left(\mathbf{u}\cdot \stackrel{̂}{\mathbf{a}}\right)\stackrel{̂}{\mathbf{a}}=\frac{1}{25}\left(4\mathbf{j}-3\mathbf{k}\right)$.
Choice (d) is incorrect
In both of the diagrams below, the vectors $\mathbf{u}$ and $\mathbf{v}$ 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 $\mathbf{u}×\mathbf{v}$ pointing?
 (1) (2)
Exactly one option must be correct)
 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).

Choice (a) is incorrect
The direction of $\mathbf{u}×\mathbf{v}$ in (1) is different from that in (2).
Choice (b) is incorrect
The direction of $\mathbf{u}×\mathbf{v}$ in (1) is different from that in (2).
Choice (c) is correct!
Choice (d) is incorrect
The direction of $\mathbf{u}×\mathbf{v}$ is such that $\mathbf{u},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathbf{v},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathbf{u}×\mathbf{v}$ form a right-hand set.
Calculate the vector cross product $\mathbf{a}×\mathbf{b}$ when $\mathbf{a}=3\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=4\mathbf{i}-\mathbf{j}$. Exactly one option must be correct)
 a) $-2\mathbf{i}-8\mathbf{j}-7\mathbf{k}$ b) $-2\mathbf{i}+8\mathbf{j}-7\mathbf{k}$ c) $2\mathbf{i}-8\mathbf{j}+7\mathbf{k}$ d) $4\mathbf{j}+2\mathbf{k}$

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