are two vectors. The positions vectors of the points A and C are respectively. Find the position vector of a point P on the line AB and a point Q on the line CD, such that PQ is perpendicular to AB and CD both.
Given,
And the position vectors
Therefore, the line passing through A and along AB will have the equation:
and the line passing through C and along CD will have equation
Now, PQ is a vector perpendicular to both AB and CD, such that Q lies on CD and P lies on AB. Thus, coordinates of P and Q will be of the form
Hence, the vector PQ will be given as
Now, since PQ is perpendicular to both, hence the dot products of AB.PQ and CD.PQ will be equal to 0.
AB. PQ = 0 and CD. PQ = 0
Solving (iii) and (iv), we get
Putting the value of in (i) we get,
Putting the value of in (ii) we get,
Hence, position vector of P and Q will be