Write the equation of the line through M_1(0,1,-2) making…

Write the equation of the line through $M_1(0,1,-2)$ making angles $60^\circ$, $45^\circ$, $120^\circ$ with the coordinate axes.

Step 1: Solution

The direction cosines are $\tfrac12,\ \tfrac{\sqrt2}2,\ -\tfrac12$. They are consistent, since $\tfrac14+\tfrac12+\tfrac14=1$. So $\vec d\parallel\{1,\sqrt2,-1\}$.

\[\frac x1=\frac{y-1}{\sqrt2}=\frac{z+2}{-1}\]

Parametric form: $x=t,\ y=1+\sqrt2\,t,\ z=-2-t$. Check: $|\vec d|=2$, so the cosines are $\tfrac12,\tfrac{\sqrt2}2,-\tfrac12$.