Equations of Lines — Question 8

Question 8 – Solution

Strategy Angle-bisector directions are the sum and difference of unit direction vectors. Here the given vectors already have equal length.

Bisector directions Since $\|\vec d_1\|=\|\vec d_2\|=5$, the sum and difference give $$\vec b_+=\left\langle 3,4,7\right\rangle,\qquad \vec b_-=\left\langle 3,-4,1\right\rangle.$$ Therefore the bisectors are $$\boxed{\vec r=\left\langle 1,-2,3\right\rangle+t\left\langle 3,4,7\right\rangle},\qquad \boxed{\vec r=\left\langle 1,-2,3\right\rangle+u\left\langle 3,-4,1\right\rangle}.$$

Acute-angle bisector Because $\vec d_1\cdot\vec d_2=12>0$, the directions form an acute angle. Their sum $\vec b_+$ lies between them and bisects that acute angle; $\vec b_-$ bisects the supplementary angle.

Perpendicularity Directly, $$\vec b_+\cdot\vec b_-=3(3)+4(-4)+7(1)=9-16+7=0.$$ Hence the two angle-bisector lines are perpendicular.

Verification Dotting $\vec b_+$ with each original direction gives the same normalized cosine, confirming equal angles.