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C4 Vectors Question help.

This is probably a really simple question but I just can't get it! Its taken from the June 2008 C4 paper, Q4:

Relative to an origin, O, the points A and B have position vectors 3i+2j+3K and i+3j+4k respectively
i) Find a vector equation for the line passing through A and B.
ii) Find the position vector of the point P on AB such that OP is perpendicular to AB.

I'm really struggling with part (ii) and could do with some help if that's ok. Thank you :smile:

Reply 1

ghostwalker

Original post by jmhoneywood1

Try searching TSR for "3i+2j+3K and i+3j+4k "; this question has popped up several times.

Reply 2

Freakonomics123

Any point on the line has position vector \overrightarrow{OP} = \begin{pmatrix} 3-2t \ 2+t \ 3+t \end{pmatrix} . You've worked out that \overrightarrow{AB} = \begin{pmatrix}-2 \ 1 \ 1 \end{pmatrix}, so you need to find such that \overrightarrow{OP} \cdot \overrightarrow{AB} = 0.

Reply 3

jmhoneywood1

Original post by Freakonomics123

Ah, thank you, that helps me solve the rest of the question. But I was just wondering (if you don't mind) how you know that OP=(3-2t)/(2-t)/(3-t) ? thank you again.
:smile:

Reply 4

The Bagel Guy

Wouldn't you just find an equation where OP.AB = 0 (using the dot product)
You know the equation will be in the order of ax + by + cz (as it passes through the origin)
And that it will intersect AB for which you have the equation

Might not have done a terrific job of explaining it and I haven't got a pen and paper to actually do it right now

Reply 5

Freakonomics123

Original post by jmhoneywood1

Ah, thank you, that helps me solve the rest of the question. But I was just wondering (if you don't mind) how you know that OP=(3-2t)/(2-t)/(3-t) ? thank you again.
:smile:

because =(3-2t)/(2-t)/(3-t) is a point on the vector (the perpendicular point) and the direction vector relative to the origin will be this value.