Answer to Question #240207 in Analytic Geometry for Appu

Question

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

x + y + z = 1 and d 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

a) if Ax + Bx +Cz =D is the equation of a plane , then the vector normal to the given plane is $n_1 =i + 2 j -k$

Equation of f2 is $3x-y+4z=2$

Therefore the normal vector $n_2= 3i - j +4k$

b) We have $n_1 =i + 2 j -k$ and $n_2= 3i - j +4k.$

Then $n_1$ and $n_2$ are perpendicular to the line of intersection. Therefore $v = n_1 * n_2$ is the perpendicular to both $n_1$ and $n_2$ and hence parallel to the plane line of intersection

$\begin{bmatrix} i & j & k \\ 1 & 2 & -1\\ 3 & -1 & 4\\ \end{bmatrix}= i(8-1)-j(4+3)+k(-1-6)=7i-7j-7k$