Q.The direction ratios of a line are $2,-3,6$. Find its direction cosines.

The direction angles alpha, beta, gamma of a non-zero vector are the angles it makes with the positive X, Y and Z axes; their cosines l=cos(alpha), m=cos(beta), n=cos(gamma) are called its direction cosines, and they always satisfy l^2+m^2+n^2=1, with l i+m j+n k being exactly the unit vector along the given direction. Any three numbers a, b, c proportional to l, m, n (that is, l/a=m/b=n/c) are called direction ratios of the same line; a line has infinitely many direction-ratio triples but only one (unsigned, i.e. up to an overall sign for the two opposite senses of the line) set of direction cosines, recovered from ratios a,b,c by dividing each by sqrt …