Oh, come on, vectors aren't all that bad. Are they?

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Question 1

Line $l_1$ has the equation $\mathbf{r}= \begin{pmatrix}1\\2\\2 \end{pmatrix} + \lambda \begin{pmatrix}4\\-3\\3\end{pmatrix}$.
Line $l_2$ has the equation $\mathbf{r}= \begin{pmatrix}5\\3\\1 \end{pmatrix} + \mu \begin{pmatrix}-12\\5\\-5\end{pmatrix}$.
Where do lines $l_1$ and $l_2$ intersect?

A

$\begin{pmatrix}5\\-1\\5\end{pmatrix}$

B

$\begin{pmatrix}-7\\8\\-4\end{pmatrix}$

C

$\begin{pmatrix}17\\-2\\8\end{pmatrix}$

D

$\begin{pmatrix}1\\2\\2\end{pmatrix}$

E

You can't fool me, they don't!

Question 2

Line $l_1$ has the equation $\mathbf{r}= \begin{pmatrix}1\\2\\2 \end{pmatrix} + \lambda \begin{pmatrix}4\\-3\\3\end{pmatrix}$.
Line $l_2$ has the equation $\mathbf{r}= \begin{pmatrix}5\\3\\3 \end{pmatrix} + \mu \begin{pmatrix}-12\\5\\-7\end{pmatrix}$.
What is the acute angle between $l_1$ and $l_2$?

A

43º

B

None of these

C

13º

D

137º

E

167º

Question 3

Point $A$ is at $\begin{pmatrix} 9\\ 0 \\ 3 \\ \end{pmatrix}$ and point $B$ is at $\begin{pmatrix} -3\\ 8 \\ -3 \\ \end{pmatrix}$. Which of the following is an equation of the line through $A$ and $B$?

Point $A$ is at $\begin{pmatrix} 9\\ 0 \\ 3 \\ \end{pmatrix}$ and point $B$ is at $\begin{pmatrix} -3\\ 8 \\ -3 \\ \end{pmatrix}$. What is the distance $AB$?

A

$244$

B

$\sqrt{244}$

C

$71$

D

$109$

E

$26$

Question 5

Point $A$ is at $\begin{pmatrix} 9\\ 0 \\ 3 \\ \end{pmatrix}$ and point $B$ is at $\begin{pmatrix} -3\\ 8 \\ -3 \\ \end{pmatrix}$. What is the vector $AB$?

A

$\begin{pmatrix}
12\\ -8 \\ 6 \\
\end{pmatrix}$

B

71º

C

$\sqrt{244}$

D

109º

E

$\begin{pmatrix}
-12 \\ 8 \\ -6 \\
\end{pmatrix}$

Question 6

Line $l_1$ has the equation $\mathbf{r}= \begin{pmatrix}1\\2\\2 \end{pmatrix} + \lambda \begin{pmatrix}4\\-3\\3\end{pmatrix}$.
Point $C$ has co-ordinates $(-1,3,0)$
Point $P$ lies on $l_1$ such that the vector $CP$ is perpendicular to $l_1$. What are the co-ordinates of $P$?

A

$(5,-1,5)$

B

$(-1, 3.5, 0.5)$

C

$(3, 0.5, 3.5)$

D

$(2,3,0)$

E

There is no such point.

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Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

In the question about the equation for the line through A and B, your TeX isn’t rendering properly.

The question about the acute angle between l1 and l2, I think the answer should be none of these since the lines don’t intersect. Then the next question asks about their intersection point and I still think they don’t intersect! Maybe I’m making a huge mistake here. I get $\lambda = -2$ and $\mu = 1$ to make the $x$ and $y$ coordinates match, but then I get $-4$ and $-6$ for the $z$ coordinate.

Also you shouldn’t say “the equation”, you should say “an equation”.

## MathbloggingAll

C4 Vectors quiz http://t.co/9bpTtnvZWr

## Joshua Zucker

In the question about the equation for the line through A and B, your TeX isn’t rendering properly.

The question about the acute angle between l1 and l2, I think the answer should be none of these since the lines don’t intersect. Then the next question asks about their intersection point and I still think they don’t intersect! Maybe I’m making a huge mistake here. I get $\lambda = -2$ and $\mu = 1$ to make the $x$ and $y$ coordinates match, but then I get $-4$ and $-6$ for the $z$ coordinate.

Also you shouldn’t say “the equation”, you should say “an equation”.