## MATH1014 Quizzes

Quiz 1: Introduction to Vectors
Question 1 Questions
Which of the following symbols represent vectors? Click on every option that does. (Zero or more options can be correct)
 a) $\mathbf{v}$ b) $-\stackrel{\to }{PQ}$ c) $2\mathbf{u}+\stackrel{\to }{AB}$ d) $-\mathbf{v}$ e) $\mathbf{u}+\mathbf{v}$ f) $2\mathbf{u}-3\mathbf{v}$

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be True.
This is the vector pointing in the opposite direction to $\stackrel{\to }{PQ}$.
There is at least one mistake.
For example, choice (c) should be True.
Both $\mathbf{u}$ and $\stackrel{\to }{AB}$ are vectors, so their sum is also a vector.
There is at least one mistake.
For example, choice (d) should be True.
The negative of a vector is a vector pointing in the opposite direction.
There is at least one mistake.
For example, choice (e) should be True.
The sum of two vectors is a vector.
There is at least one mistake.
For example, choice (f) should be True.
Both $2\mathbf{u}$ and $-3\mathbf{v}$ are vectors and their sum is also a vector.
Correct!
1. True
2. True This is the vector pointing in the opposite direction to $\stackrel{\to }{PQ}$.
3. True Both $\mathbf{u}$ and $\stackrel{\to }{AB}$ are vectors, so their sum is also a vector.
4. True The negative of a vector is a vector pointing in the opposite direction.
5. True The sum of two vectors is a vector.
6. True Both $2\mathbf{u}$ and $-3\mathbf{v}$ are vectors and their sum is also a vector.

Correct!
Vectors are equal when they have the same direction and length; their position in space does not matter.
Remember that a vector is specified by its direction and magnitude, so that the two arrows of equal length pointing to the west represent the same vector, while the three arrows of equal length pointing to the north-east also represent the same vector.
How many different vectors are drawn here ?

Correct!
All of the vectors are different. However, some of these vectors are scalar multiples of each other; for example,

Vectors are equal when they have the same direction and length; their position in space does not matter.
Express the vector $\mathbf{u}$ in terms of $\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b},\phantom{\rule{0.3em}{0ex}}\mathbf{c}$.
Exactly one option must be correct)
 a) $-\mathbf{a}+\mathbf{b}+\mathbf{c}$ b) $\mathbf{a}-\mathbf{b}+\mathbf{c}$ c) $\mathbf{a}+\mathbf{b}+\mathbf{c}$ d) $-\mathbf{a}-\mathbf{b}-\mathbf{c}$

Choice (a) is incorrect
Trace out the vector $\mathbf{u}$ starting at the tail and moving along the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ until you reach the head of $\mathbf{u}$.
Choice (b) is incorrect
Trace out the vector $\mathbf{u}$ starting at the tail and moving along the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ until you reach the head of $\mathbf{u}$.
Choice (c) is incorrect
Trace out the vector $\mathbf{u}$ starting at the tail and moving along the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ until you reach the head of $\mathbf{u}$.
Choice (d) is correct!
Starting at the tail of $\mathbf{u}$, we see that $\mathbf{u}=-\mathbf{a}-\mathbf{b}-\mathbf{c}$.
Express the vector $\mathbf{u}$ in terms of $\mathbf{a},\phantom{\rule{0.3em}{0ex}}\mathbf{b},\phantom{\rule{0.3em}{0ex}}\mathbf{c}$.
Exactly one option must be correct)
 a) $-\mathbf{a}+\mathbf{b}+\mathbf{c}$ b) $\mathbf{a}-\mathbf{b}-\mathbf{c}$ c) $\mathbf{a}+\mathbf{b}+\mathbf{c}$ d) $-\mathbf{a}-\mathbf{b}-\mathbf{c}$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
If $\stackrel{\to }{OP}=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill -2\hfill \end{array}\right],$ $\stackrel{\to }{OQ}=\left[\begin{array}{c}\hfill -2\hfill \\ \hfill 1\hfill \end{array}\right]$and $\stackrel{\to }{QP}=\stackrel{\to }{AB},$ where $B=\left(3,5\right),$ find $A$. Exactly one option must be correct)
 a) $A=\left(2,-8\right)$ b) $A=\left(-2,-8\right)$ c) $A=\left(-2,8\right)$ d) $A=\left(-1,8\right)$ e) $A=\left(-2,7\right)$

Choice (a) is incorrect
First find $\stackrel{\to }{QP}$. Then let $A=\left(a,b\right)$ so that $\stackrel{\to }{QP}=\stackrel{\to }{AB}=\left[\begin{array}{c}\hfill 3-a\hfill \\ \hfill 5-b\hfill \end{array}\right]$, and solve for $a$ and $b$.
Choice (b) is incorrect
First find $\stackrel{\to }{QP}$. Then let $A=\left(a,b\right)$ so that $\stackrel{\to }{QP}=\stackrel{\to }{AB}=\left[\begin{array}{c}\hfill 3-a\hfill \\ \hfill 5-b\hfill \end{array}\right]$, and solve for $a$ and $b$.
Choice (c) is correct!
Choice (d) is incorrect
First find $\stackrel{\to }{QP}$. Then let $A=\left(a,b\right)$ so that $\stackrel{\to }{QP}=\stackrel{\to }{AB}=\left[\begin{array}{c}\hfill 3-a\hfill \\ \hfill 5-b\hfill \end{array}\right]$, and solve for $a$ and $b$.
Choice (e) is incorrect
First find $\stackrel{\to }{QP}$. Then let $A=\left(a,b\right)$ so that $\stackrel{\to }{QP}=\stackrel{\to }{AB}=\left[\begin{array}{c}\hfill 3-a\hfill \\ \hfill 5-b\hfill \end{array}\right]$, and solve for $a$ and $b$.
If $A=\left(3,-1\right),\phantom{\rule{0.3em}{0ex}}B=\left(3,5\right)$ and $C=\left(-2,0\right)$, find $P$ such that $\stackrel{\to }{OP}=\stackrel{\to }{AB}+2\stackrel{\to }{BC}$. Exactly one option must be correct)
 a) $P=\left(10,7\right)$ b) $P=\left(10,14\right)$ c) $P=\left(-10,-4\right)$ d) $P=\left(-20,-26\right)$

Choice (a) is incorrect
Find $\stackrel{\to }{AB}$ and $\stackrel{\to }{BC}$ first. Then find $2\stackrel{\to }{BC}$ and hence $\stackrel{\to }{OP}.$
Choice (b) is incorrect
Find $\stackrel{\to }{AB}$ and $\stackrel{\to }{BC}$ first. Then find $2\stackrel{\to }{BC}$ and hence $\stackrel{\to }{OP}.$
Choice (c) is correct!
We have $\stackrel{\to }{OP}=\stackrel{\to }{AB}+2\stackrel{\to }{BC}=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 6\hfill \end{array}\right]+2\left[\begin{array}{c}\hfill -5\hfill \\ \hfill -5\hfill \end{array}\right]=\left[\begin{array}{c}\hfill -10\hfill \\ \hfill -4\hfill \end{array}\right].$ Hence $P=\left(-10,-4\right).$
Choice (d) is incorrect
Find $\stackrel{\to }{AB}$ and $\stackrel{\to }{BC}$ first. Then find $2\stackrel{\to }{BC}$ and hence $\stackrel{\to }{OP}.$
Relative to the origin, point $P$ has position vector $\mathbf{u}$ and $Q$ has position vector $\mathbf{v}$. What is $\stackrel{\to }{QP}$ ? 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!
$\stackrel{\to }{QP}=\stackrel{\to }{OP}-\stackrel{\to }{OQ}=\mathbf{u}-\mathbf{v}$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
In 3D space, vectors $\mathbf{u}$ and $\mathbf{v}$ are defined as follows: $\mathbf{u}=\left[3,5,-1\right],\phantom{\rule{2em}{0ex}}\mathbf{v}=\left[1,4,7\right].$ If the tail of vector $\mathbf{u}+\mathbf{v}$ is translated to the point $P=\left(6,-2,1\right)$, which point corresponds to the head $Q$ of $\mathbf{u}+\mathbf{v}$? Exactly one option must be correct)
 a) $\left(4,9,6\right)$ b) $\left(10,7,7\right)$ c) $\left(0,0,0\right)$ d) $\left(-2,3,5\right)$ e) $\left(2,-11-5\right)$

Choice (a) is incorrect
Choice (b) is correct!
We find that $\mathbf{u}+\mathbf{v}=\left[4,9,6\right]$ and if the coordinates of $Q$ are $\left(a,b,c\right)$ then $\mathbf{u}+\mathbf{v}=\stackrel{\to }{PQ}=\left[a-6,b+2,c-1\right]$. This gives the result.
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which vector is obtained when the vector expression $5\left(\mathbf{v}-2\mathbf{u}\right)-3\left(\mathbf{v}-4\mathbf{w}\right)+3\left(\mathbf{u}-\mathbf{v}+\mathbf{w}\right)$ is simplified? Exactly one option must be correct)
 a) $-7\mathbf{u}-\mathbf{v}+9\mathbf{w}$ b) $-10\mathbf{u}-\mathbf{v}+15\mathbf{w}$ c) $-7\mathbf{u}+\mathbf{v}+15\mathbf{w}$ d) $-7\mathbf{u}-\mathbf{v}+15\mathbf{w}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!