18. (a) The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are defined by \(\mathbf { u } = 2 \mathbf { i } - 3 \mathbf { j } , \mathbf { v } = - 4 \mathbf { i } + 5 \mathbf { j }\).
- Find the vector \(4 \mathbf { u } - 3 \mathbf { v }\).
- The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are the position vectors of the points \(U\) and \(V\), respectively. Find the length of the line UV.
(b) Two villages \(A\) and \(B\) are 40 km apart on a long straight road passing through a desert. The position vectors of \(A\) and \(B\) are denoted by \(\mathbf { a }\) and \(\mathbf { b }\), respectively. - Village \(C\) lies on the road between \(A\) and \(B\) at a distance 4 km from \(B\). Find the position vector of \(C\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\).
- Village \(D\) has position vector \(\frac { 2 } { 9 } \mathbf { a } + \frac { 5 } { 9 } \mathbf { b }\). Explain why village \(D\) cannot possibly be on the straight road passing through \(A\) and \(B\).