Question

If the direction cosines of a line are l, m and n then which of the following is true?

• A.
• B.
• C.
• D.

Answer 1

The correct option is D


et vector OP = r, r is the position vector of point P(x,y,z)
$\sqrt{x^2+y^2+z^2}=|r|$
$x^2+y^2+z^2=(|r|{)}^2$.....(1)
And we know that if a line segment of magnitude “r” makes angles $\alpha$ $\beta$ and $\gamma$ with x,y and z axes
then x=r $cos\alpha$,y=r $cos\beta$ $cos\gamma$ are nothing but the direction cosines, which are given as l, m and n, so-
x=l.r,y=m.r and z=n.r
So, we’ll have - (using (1))
$(l.r{)}^2+(m.r{)}^2+(n.r{)}^2=(|r|{)}^2$
or $r^2(l^2+m^2+n^2)=r^2$
or $l^2+m^2+n^2=1$