Let L_1: x-11=y-2-1=z-12 and L_2: x+1-1=y-22=z1 be two lines.

Question

Let $L_1:\ \dfrac{x-1}{1}=\dfrac{y-2}{-1}=\dfrac{z-1}{2}$ and $L_2:\ \dfrac{x+1}{-1}=\dfrac{y-2}{2}=\dfrac{z}{1}$ be two lines. Let $L_3$ be a line passing through the point $(1,2,1)$... (through a point on $L_1$) and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $L_1$ and meets it, and $(\alpha,\beta,\gamma)$ is a point on $L_3$, then $|5\alpha-11\beta-8\gamma|$ equals:

Accepted Answer

$L_1$: point $P_1=(1,2,1)$, direction $\vec u=(1,-1,2)$. $L_2$: point $P_2=(-1,2,0)$, direction $\vec v=(-1,2,1)$.
$$\vec d=\vec u\times\vec v=(-5,-3,1).$$
A general point on $L_1$ is $A(t)=(1+t,\ 2-t,\ 1+2t)$, and $(\alpha,\beta,\gamma)=A(t)+s\vec d$:
$$\alpha=1+t-5s,\quad \beta=2-t-3s,\quad \gamma=1+2t+s.$$
$$5\alpha-11\beta-8\gamma=(5-22-8)+(5t+11t-16t)+(-25s+33s-8s)=-25.$$
$$|5\alpha-11\beta-8\gamma|=25.$$

Solutions · 1