# Vectors (Perpendicular vectors) problem

#### kandyfloss

Position vec of points A and B with respect to origin O are 2i + 4j and 6i + 7 J

question:- position vector of point C is 3i-4j. find the vector equation for the line OC and show OC is right angle to AB.
and also - Find the position vector of point of intersection of OC and Ab and hence find the perpendicular distance from O to AB.

i did r= 0 (because its the origin) + u(3i-4j) , where u is a scalar quantity..i don't know how to show its right angles with AB and also the following parts.

#### Unknown008

If two vectors are perpendicular to each other, then their dot product is equal to 0.

So, find the vector AB, then find the dot product of the vectors AB and OC. If this gives zero, then it means that AB and OC vectors are perpendicular to each other.

For the second part, since you just showed that vector OC is perpencicular to vector AB, you already have showed that the shortest distance from O to AB 'lies' on the line OC. So, find the point where line OC and AB meet, and from there, you can find the distance from their point of intersection to the origin, which is also the shortest distance from O to line AB.

I hope I didn't confuse you too much (Itwasntme)

#### kandyfloss

Can you show me with the values given?
I don't understand how to dot to vector equations of a line in the form r= ai + bj + L(ci + dj)

#### Unknown008

In the dot product of vectors, you only take into consideration the direction vectors.

For the first part, you do:

(4i + 3j)(3i - 4j) = (4*3) + (3*-4)

or:

$$\displaystyle \left(\begin{array}{c}4\\3\end{array}\right). \left(\begin{array}{c}3\\-4\end{array}\right)= (4 \times 3) + (3 \times -4)$$

Phew, this LaTeX is more confusing than the one I use to work with...

• kandyfloss