Stuck on a vector question m1

here is my working. I am a bit confused because directions are given in the question so can you use the isoceles triangle to do work it out? I really don't see how this works. Thanks.

It should be $\mathbf{r} = -2\mathbf{i}$ for the position vector of the ship at 10am.

Have another go with that corrected.

Oh I got that as I have done the question again today then I amnot sure what to do.

Try sketching some axes and roughly plot the positions of the lighthouse and the ship at the two times. Then form some triangles and label as many lengths and angles as you can.

Please post your diagram if you get stuck.

I am stuck on the south west bit. This is what I have done so far. Thanks.

I am stuck on the south west bit. This is what I have done so far. Thanks.

You didn't do what I suggested:

First sketch some axes. Then plot the position of tthe ship at 10am. It's position is $-2\mathbf{i}$ so plot the coordinate $(-2,0)$.

L is due west of $(-2,0)$ so plot L somewhere on the x-axis to the left of $(-2,0)$.

Now plot the position of tthe ship at 10.30am. It's position is $-5\mathbf{i}+4\mathbf{j}$ so roughly plot the coordinate $(-5,4)$.

You are told that L is southwest of this position so draw a line from $(-5,4)$ to L which forms a 45 degree angle and label this angle.

Try to use this diagram to find the coordinates of L. Post your complete diagram if you get stuck.

Perhaps if you drew your triangle so that the lighthouse and the ship at 10am were on the same horizontal line (due west) and then the other one in a relative position and put in the 45 degree angle, you might see something.

sohow will I be able to work out what l is at? I think this is the right diagram.. I know that between 10 and 10:30 the difference in I is 3 but I am not sure what the other two angles would be. Thanks.

You can use the fact that the top vertex has coordinates (-5,4) to get some lengths in your diagram. The horizontal length from (-2,0) to (-5,4) is 3 which is where the 3 in the diagram below has come from:



Can you now work out the length from (-2,0) to L?

Part c) wants you to find the position vector of L which is just the coordinates of L written as a vector.

You have also calculated the distance from the position of the ship at 10:30 to the lighthouse. Your working is correct but it is not needed for this question.

I have just worked it out but now I don't get which length we want to find out as I thought it was the 4root 2 one... Thanks.

Also, a couple of tips to speed up your trig working:

The base of the left right-angled triangle must be 4 because it's isosceles and its height is 4. There's no need to use the sine rule.
And to find the 4sqrt(2) length you could have used Pythagoras with 4 and 4.

Oh okay. So if it's asking for the position vector of L from ship it would be 7i. But if it wants position vector of L from s at 10:30, 4√2 would be the answer? Thanks.

Thanks. I actually worked out 4√2 first that's why. The right answer is -9i and I think it's just -2-7 . The question asks for position vector but it doesn't say from the origin so how do you know whether it's from the ship or origin? Do you have to assume it's from the origin?
Also in the mark scheme the working is simply -5-4i so I don't get how that works...

The position vector of a point by definition means the vector from the origin to the point.

It's actually $-7\mathbf{i}$. Does this make sense?

"The position vector of L from S" - this doesn't really make sense here. A position vector of a point is a vector that goes to the point from some origin. But the origin has already been fixed in the question so you can't relocate it to S.

Also, 4√2 is the distance of L from S at 10:30. 4√2 is not a vector.

If instead you wanted the 'Vector of L from S at 10:30' then it is 4 units down and 4 units right so the vector would be $-4\mathbf{i} - 4\mathbf{j}$. And this vector has magnitude 4√2.