Find the direction cosines of the vector joining the points `A(1,2,-3)` and `B(-1,-2,1)` directed from A to B.

Explanation

The given points are `A(1,2,-3) and B(-1,-2,1)`
i.e., `x_(1)=1,y_(1)=2,z_(1)=-3` and `x_(2)=-1,z_(2)=1`
Vector `AB=(x_(2)-x_(1))hat(i)+(y_(2)-y_(1))hat(j)+(z_(2)-z_(1))hat(k)`.
`=(-1-1)hat(i)+(-2-2)hat(j)+[1-(-3)]hat(k)=-2hat(i)-4hat(j)+4hat(k)`.
Comparing with `X=xhat(i)+yhat(j)+zhat(k)`, we get `x=-2,y=-4,z=4`
Now , magnitude `|AB|=sqrt(x^2+y^2a+z^2)=sqrt((-2)^2+(-4)^2+4^2)=sqrt(4+16+16)=sqrt(36)=6`.
Direction cosines of a vector `overline(X)=x hat(i)+yhat(j)+zhat(k)`, are `(x)/(|vecX|)(y)/(|vecX|),(z)/(|vecX|)`
`therefore` Direction cosines of AB are `(-2)/(6),(-4)/(6),(4)/(6) or -(1)/(2),-(2)/(3),(2)/(3)`.