Question

(1) 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 ny and n₂ be the normal vectors to the two given planes. Without actual computation, describe the relationship between d and n₁ x n₂.

Answer 1

a) $n_1=\langle 1, 1, 1\rangle, n_2=\langle -1, 2, 10\rangle$

$$n_1\times n_2=\begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ -1 & 2 & 10 \\ \end{vmatrix}$$

$$=i\begin{vmatrix} 1 & 1 \\ 2 & 10 \end{vmatrix}-j\begin{vmatrix} 1 & 1 \\ -1 & 10 \end{vmatrix}+k\begin{vmatrix} 1 & 1 \\ -1 & 2 \end{vmatrix}$$

$$=8i-11j+3k$$

$$d=\langle 8, -11, 3\rangle$$

$$x+y+z= 1$$ $$-x+2y + 10z =2$$

$$x+y+z= 1$$ $$3y+11z= 3$$

$x_1=0, y_1=1, z_1=0$

$x_2=8, y_2=-10, z_2=3$

$$d=\langle x_2-x_1, y_2-y_1, z_2-z_1\rangle$$

$$d=\langle 8-0, -10-1, 3-0\rangle$$

$$d=\langle 8, -11, 3\rangle$$

b)

$$d=n_1\times n_2$$

$$d\perp n_1, d\perp n_2$$