9709 P32 - Jun 2022 - Q9 - 10 marks
2176
The lines \(l\) and \(m\) have vector equations \[\mathbf r=-\mathbf i+3\mathbf j+4\mathbf k+\lambda(2\mathbf i-\mathbf j-\mathbf k)\] and \[\mathbf r=5\mathbf i+4\mathbf j+3\mathbf k+\mu(a\mathbf i+b\mathbf j+\mathbf k)\] respectively, where \(a\) and \(b\) are constants.
(a) Given that \(l\) and \(m\) intersect, show that \(2b-a=4\). [4]
(b) Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\). [4]
(c) When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\). [2]
