91Ó°ÊÓ

Find the direction cosines of the vectors whose direction ratios are \((3,4,5)\) and \((1,2,-3)\). Hence find the angle between the two vectors.

Find the direction cosines of the vector joining the two points \((4,2,2)\) and \((7,6,14)\).

If \(\mathbf{a}=5 \mathbf{i}+4 \mathbf{j}+2 \mathbf{k}, \mathbf{b}=4 \mathbf{i}-5 \mathbf{j}+3 \mathbf{k}\) and \(\mathbf{c}=2 \boldsymbol{i}-\mathbf{j}-2 \mathbf{k}\), where \(\mathbf{1}, \mathbf{j}, \mathbf{k}\) are the unit vectors, determine: (a) the value of a b and the angle between the vectors a and \(\mathbf{b}\) (b) the magnitude and the direction cosines of the product vector \((\mathbf{a} \times \mathbf{b})\) and also the angle which this product vector makes with the vector \(c\).