Difficult Vector Questions?

**Kayla_N** · January 22nd 2010, 11:43 AM

I'm having trouble with these problems..matter of fact I don't even know where to start..please help thanks.

1. Find the distance of the point

from the line through

which points in the direction of

.

2. Find the vector

which makes an angle of 60 degrees with the vector

and which is of the same length as

, and is counterclockwise to

.

3. You start walking from a point with coordinates (1, 1) and arrived at the point with coordinates (2, 5). If you began walking in the direction of the vector

and you change direction only once, when you make a turn at a right angle, what are the coordinates of the point where you make the turn?

I figured out 2 and 3 already. I still need help on 1. Thanks

**Sudharaka** · January 22nd 2010, 05:34 PM

Dear Kayla N,

$A\equiv{(8,-4)}\mbox{ and }B\equiv{(-8,-1)}$

$AB=-16i+3j$

$\mid{AB}\mid=\sqrt{(16^2+3^2)}\approx{16.28}$

$(-16i+3j).(-7i+4j)=\mid{(-16i+3j)}\parallel{(-7i+4j)}\mid{Cos\theta}$ ; $\theta\mbox{ is the acute angle between AB and (-7i+4j)}$

$\theta\approx{19.13^o}$

Therefore perpendicular distance to the line from the point (8,-4) = $\mid{AB}\mid{Sin19.13^o}\approx{5.33}$

Hope this helps.

**Grandad** · January 22nd 2010, 10:23 PM

Hello Kayla_N

Originally Posted by Kayla_N

...
1. Find the distance of the point

from the line through

which points in the direction of

.
...I still need help on 1. Thanks

The line in the direction $-7\textbf{i}+4\textbf{j}$ has gradient $-\tfrac47$ , so its equation is:

$y+1=-\tfrac47(x+8)$

i.e. $7y+1=-4x-32$

i.e. $4x+7y+33=0$

Now use the formula:

Distance of $(x_1,y_1)$ from $ax+by+c=0 =\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$

to get:

$\frac{|32-28+33|}{\sqrt{16+49}}\approx 4.59$

if my working is correct.

Grandad

**Sudharaka** · January 22nd 2010, 11:40 PM

Dear Grandad,

In your calculation there is a slight error. Instead of $7y+1=-4x-32$ it should be $7y+7=-4x-32$ . If you proceed you would get the same result that I had achieved.

Thank you.

**Grandad** · January 23rd 2010, 05:42 AM

Thank you, Sudharaka.

Originally Posted by Sudharaka

Dear Grandad,

In your calculation there is a slight error. Instead of $7y+1=-4x-32$ it should be $7y+7=-4x-32$ . If you proceed you would get the same result that I had achieved.

Thank you.

I was obviously in too much of a hurry when I first did this!

My revised answer, then, is: