Equations of Lines — Question 4

Question 4

The lines $$L_1:\vec r=\left\langle 1,2,-1\right\rangle+s\left\langle 2,-1,2\right\rangle,\qquad L_2:\vec r=\left\langle 5,0,3\right\rangle+t\left\langle 1,2,-2\right\rangle$$ intersect. Find their intersection and the acute angle between them.

Question 4 – Solution

Strategy Solve the coordinate equations for the intersection; use direction vectors for the angle.

Intersection Equating coordinates gives $1+2s=5+t$, $2-s=2t$, and $-1+2s=3-2t$. The solution is $s=2,t=0$, so $\boxed{Q=(5,0,3)}$.

Angle With $\vec d_1=\left\langle 2,-1,2\right\rangle$ and $\vec d_2=\left\langle 1,2,-2\right\rangle$, $$|\vec d_1\cdot\vec d_2|=4,\quad \|\vec d_1\|=3,\quad\|\vec d_2\|=3.$$ Hence $\boxed{\theta=\cos^{-1}(4/9)\approx 63.6^\circ}$.

Verification Substituting $s=2$ and $t=0$ gives the same point on both lines.