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C4 Vectors

The points A and B have position vectors:

OA=7i+4j−3k[br][br]OB=11i+8j+5k\boldsymbol{OA} = 7\boldsymbol{i} + 4\boldsymbol{j} - 3\boldsymbol{k}[br][br]\boldsymbol{OB} = 11\boldsymbol{i} + 8\boldsymbol{j} + 5\boldsymbol{k}

Where O is the origin. The point P lies on AB and is such that AP:PB=1:3AP : PB = 1 : 3.

a) Find an expression for the vector AB\boldsymbol{AB} in the terms of i,j,k\boldsymbol{i, j, k}.

Spoiler



b) Find an expression for the position vector OP\boldsymbol{OP} in the terms of i,j,k\boldsymbol{i, j, k}.

This is where I have no idea; I (vaguely) remember about some trick to do with ratios and vectors but I have no clue, any idea?
Reply 1
Avatar for Adjective
Adjective
12
Find either the vector AP or BP. Then OP = OA + AP = OB + BP.
Reply 2
Avatar for etp
etp
OP
11
How can I find those vectors when I don't know what P is?

or is it just a quater of AB so that P is i+j+2k\boldsymbol{i} + \boldsymbol{j} + 2\boldsymbol{k}?
Reply 3
Avatar for etp
etp
OP
11
This is a WJEC question :wink:
Reply 4
Avatar for IChem
IChem
11
3lliott
How can I find those vectors when I don't know what P is?

or is it just a quater of AB so that P is i+j+2k\boldsymbol{i} + \boldsymbol{j} + 2\boldsymbol{k}?


Yes. Now draw a diagram and it will help you to see how to get the vector OP
Reply 5
Avatar for Adjective
Adjective
12
3lliott

or is it just a quater of AB so that P is i+j+2k\boldsymbol{i} + \boldsymbol{j} + 2\boldsymbol{k}?


Yes, exactly. Though I hope you mean ...so that AP is... :wink:
Reply 6
Avatar for 01perryd
01perryd
I think you need the line equation. then work out the length of the vector using pythagoras and use the ratio given
Reply 7
Avatar for etp
etp
OP
11
Nah, I just didn't realize OP=OA+AP\boldsymbol{OP = OA + AP}, doh!
Draw a diagram for line AB. P lies on this line such that AB is split into the ratio 1:3. So how far along this line is AP?

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