## MATH1002 Quizzes

Quiz 2: Unit vectors
Question 1 Questions
Let $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ be unit vectors in the $x$, $y$ and $z$ directions, respectively. Suppose that $\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k}$. What is the magnitude of the vector $\mathbf{a}$? Exactly one option must be correct)
 a) 4 b) $\sqrt{12}$ c) $\sqrt{14}$ d) 14

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$|\mathbf{a}|=\sqrt{{3}^{2}+{\left(-1\right)}^{2}+{2}^{2}}=\sqrt{9+1+4}=\sqrt{14}.$
Choice (d) is incorrect
Suppose that $\mathbf{a}=2\mathbf{i}+\mathbf{j}$, $\mathbf{b}=\mathbf{i}-4\mathbf{j}+\mathbf{k}$, and $\mathbf{c}=\mathbf{j}+\mathbf{k}$. What is the magnitude of the vector $2\mathbf{a}-\mathbf{b}+\mathbf{c}$ ? Exactly one option must be correct)
 a) $\sqrt{58}$ b) $10$ c) $2\sqrt{5}-\sqrt{18}+\sqrt{2}$ d) $\sqrt{30}$

Choice (a) is correct!
We have $2\mathbf{a}-\mathbf{b}+\mathbf{c}=3\mathbf{i}+7\mathbf{j}$, and so $|2\mathbf{a}-\mathbf{b}+\mathbf{c}|=\sqrt{{3}^{2}+{7}^{2}}=\sqrt{58}$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Which of the of 3-dimension coordinates axes below are right–handed coordinate systems? (The arrows indicate the positive direction on each axis.) (Zero or more options can be correct)
 a) b) c) d) e)

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.
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.
Given that $\mathbf{u}=\sqrt{3}\mathbf{i}-2\mathbf{j}+3\mathbf{k}$, find the unit vector in the opposite direction to $\mathbf{u}$. Exactly one option must be correct)
 a) $\frac{\sqrt{3}}{4}\mathbf{i}-\frac{1}{2}\mathbf{j}+\frac{3}{4}\mathbf{k}$ b) $\mathbf{i}+\mathbf{j}+\mathbf{k}$ c) $-\frac{\sqrt{3}}{4}\mathbf{i}+\frac{1}{2}\mathbf{j}-\frac{3}{4}\mathbf{k}$ d) $\frac{\sqrt{3}}{\sqrt{2}}\mathbf{i}-\frac{2}{\sqrt{2}}\mathbf{j}+\frac{3}{\sqrt{2}}\mathbf{k}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
First, find the unit vector in the same direction as $\mathbf{u}$. This requires knowing the magnitude of $\mathbf{u}$, which is $|\mathbf{u}|=\sqrt{3+4+9}=4.$ Then $\frac{1}{4}\mathbf{u}$ is the unit vector in the same direction as $\mathbf{u}$, so its negative, $-\frac{1}{4}\mathbf{u}$, is the answer.
Choice (d) is incorrect
Given that $\mathbf{u}=3\mathbf{i}+2\mathbf{j}-\mathbf{k}$, $\mathbf{v}=\mathbf{i}-3\mathbf{j}+2\mathbf{k}$, find the unit vector in the direction of $\mathbf{u}-2\mathbf{v}$. Exactly one option must be correct)
 a) $\frac{1}{3\sqrt{10}}\left(\mathbf{i}+8\mathbf{j}-5\mathbf{k}\right)$ b) $\frac{1}{3\sqrt{14}}\left(\mathbf{i}+8\mathbf{j}-5\mathbf{k}\right)$ c) $\frac{1}{\sqrt{14}}\left(\mathbf{i}+8\mathbf{j}-5\mathbf{k}\right)$ d) $\frac{1}{\sqrt{14}}\left(2\mathbf{i}+5\mathbf{j}+\mathbf{k}\right)$

Choice (a) is correct!
$\mathbf{u}-2\mathbf{v}=\mathbf{i}+8\mathbf{j}-5\mathbf{k}$.
$|\mathbf{u}-2\mathbf{v}|=\sqrt{1+64+25}=\sqrt{90}=3\sqrt{10}$
The unit vector in the direction of $\mathbf{u}-2\mathbf{v}$ is therefore
$\frac{\mathbf{u}-2\mathbf{v}}{|\mathbf{u}-2\mathbf{v}|}=\frac{1}{3\sqrt{10}}\left(\mathbf{i}+8\mathbf{j}-5\mathbf{k}\right).$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Find the polar angle (in radians) of $-2\mathbf{i}-\mathbf{j}$. Exactly one option must be correct)
 a) 0.463 b) 3.605 c) 2.677 d) $-0.463$

Choice (a) is incorrect
Choice (b) is correct!
If we set $\stackrel{⃗}{OP}=-2\mathbf{i}-\mathbf{j}$, then the point $P$ is in the third quadrant. Hence if $\theta$ is a polar angle of $\stackrel{⃗}{OP}$, then $tan\theta =\frac{1}{2}$, and so $\theta =\pi +{tan}^{-1}\frac{1}{2}\approx 3.605$ radians.
Choice (c) is incorrect
Choice (d) is incorrect
Find the polar angle (in radians) of $-\sqrt{3}\mathbf{i}+\mathbf{j}$. Exactly one option must be correct)
 a) $\frac{\pi }{6}$ b) $-\frac{\pi }{6}$ c) $\frac{5\pi }{6}$ d) $-\frac{5\pi }{6}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$tan\theta =-\frac{1}{\sqrt{3}}$ in the second quadrant so $\theta =\frac{5\pi }{6}$.
Choice (d) is incorrect
Relative to the origin, point $P$ has position vector $\mathbf{u}$ and $Q$ has position vector $\mathbf{v}$. What is $\stackrel{⃗}{QP}$ ? Exactly one option must be correct)
 a) $\mathbf{u}-\mathbf{v}$ b) $\mathbf{v}-\mathbf{u}$ c) $\mathbf{u}+\mathbf{v}$ d) $-\mathbf{u}-\mathbf{v}$

Choice (a) is correct!
$\stackrel{⃗}{QP}=\stackrel{⃗}{OP}-\stackrel{⃗}{OQ}=\mathbf{u}-\mathbf{v}$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Relative to the origin, point $P$ has position vector $3\mathbf{i}+\mathbf{j}-4\mathbf{k}$ and $Q$ has position vector $\mathbf{i}-\mathbf{j}+3\mathbf{k}$. What is $\stackrel{⃗}{QP}$ ? Exactly one option must be correct)
 a) $2\mathbf{i}+2\mathbf{j}-7\mathbf{k}$ b) $2\mathbf{i}-7\mathbf{k}$ c) $3\mathbf{i}-\mathbf{j}-12\mathbf{k}$ d) $-2\mathbf{i}-2\mathbf{j}+7\mathbf{k}$

Choice (a) is correct!
Since $\stackrel{⃗}{QP}=\stackrel{⃗}{OP}-\stackrel{⃗}{OQ}$, we have $\stackrel{⃗}{QP}=3\mathbf{i}+\mathbf{j}-4\mathbf{k}-\left(\mathbf{i}-\mathbf{j}+3\mathbf{k}\right)=2\mathbf{i}+2\mathbf{j}-7\mathbf{k}$.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A boat sails 5 km south-east then 3 km due west. Approximately how far from its starting position is it now ? Exactly one option must be correct)
 a) 8 km b) 4.2 km c) 7.4 km d) 3.6 km

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) 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.
The required distance from the starting position is then $|\stackrel{⃗}{OQ}|$. Now as $\stackrel{⃗}{OP}=\frac{5}{\sqrt{2}}\mathbf{i}-\frac{5}{\sqrt{2}}\mathbf{j}$, and $\stackrel{⃗}{PQ}=-3\mathbf{i}$, we have $\stackrel{⃗}{OQ}=\stackrel{⃗}{OP}+\stackrel{⃗}{PQ}=\left(\frac{5}{\sqrt{2}}-3\right)\mathbf{i}-\frac{5}{\sqrt{2}}\mathbf{j}.$ The magnitude of this vector is the required distance, namely $\sqrt{\frac{25}{2}+9-\frac{30}{\sqrt{2}}+\frac{25}{2}}\approx 3.6km.$