Question

a) Give parametric equation (point-direction form) of the line which lies on both of the planes:

x + y + z = 1 and −x + 2y + 10z = 2. What is the direction d of this line?

b) Let n1 and n2 be the normal vectors to the two given planes. Without actual computation,

describe the relationship between d and n1 × n2.

Answer 1

Let $y=0$ , then:

$11z=3\implies z=3/11$

$x=8/11$

The direction vector of this line:

$\overline{d}=\overline{n}_1\times \overline{n}_2=\begin{vmatrix} i & j&k \\ 1 & 1&1\\ -1&2&10 \end{vmatrix}=8i-11j+3k$

The equation of the line:

$\frac{x-8/11}{8}=\frac{y}{-11}=\frac{z-3/11}{3}$

Parametric equation of the line:

$\begin{cases} x=p_1t+x_0 \\ y=p_2t+y_0\\ z=p_3t+z_0 \end{cases}$

$\begin{cases} x=8t+8/11 \\ y=-11t\\ z=3t+3/11 \end{cases}$

$|\overline{d}|=|\overline{n}_1\times \overline{n}_2|=\sqrt{8^2+11^2+3^2}=\sqrt{194}$

direction of the line:

$cos\alpha=8/\sqrt{194}$

$cos\beta=-11/\sqrt{194}$

$cos\gamma=3/\sqrt{194}$