Question

A line makes angles of measures 45° and 60° with positive direction of y and z axes respectively. Find the d.c.s. of the line and also find the vector of  magnitude 5 along the direction of line.

Answer 1

Let α, β, γ be the direction angles of the line

 β=45°, γ=60°

`m=cosbeta=cos45^@=1/sqrt2 and n=cos gamma=cos 60^@=1/2`

Since, `cos^2alpha+cos^2beta+cos^2gamma=1`

`therefore cos^2alpha+(1/sqrt(2))^2+(1/2)^2=1`

`therefore cos^2alpha=1-1/2-1/4=1/4`

`therefore cos alpha=+-1/2`

i.e `l=+-1/2,m=1/sqrt2,n=1/2`

The unit vectors along the direction of line are

`hatu=lhati+mhatj+nhatk`

`=+-1/2hati+1/sqrt2hatj+1/2hatk`

∴ The vectors of magnitude 5 are

`5(1/2hati+1/sqrt2hatj+1/2hatk) and 5(-1/2hati+1/sqrt2hatj+1/2hatk)`