find the components of a vector of length 44 that lies on the line of intersection of the planes below...?

Question

determine the components of a vector of length 44 that lies on the line of intersection of the planes with equations 3x-4y+9z=0, and 2y-9z=0

Answer 1

Determine the components of a vector of length 44 that lies on the line of intersection of the planes with equations
3x - 4y + 9z = 0, and 2y - 9z = 0.

The directional vector v of the line of intersection of the two planes will be normal to the normal vectors of the two given planes. The normal vectors of the two given planes are:

n1 = <3, -4, 9>
n2 = <0, 2, -9>

v = n1 X n2 = <3, -4, 9> X <0, 2, -9> = <18, 27, 6>

The magnitude of v is:

|| v || = √(18² + 27² + 6²) = √(324 + 729 + 36) = √1089 = 33

We want a magnitude of 44. Multiply the vector by
44/33 = 4/3.

v = (4/3)<18, 27, 6> = <24, 36, 8>