calculus helppp

Question

Consider the planes 3𝑥+4𝑦+5𝑧=13x+4y+5z=1 and 3𝑥+5𝑧=0.3x+5z=0.

(A) Find the unique point P on the y-axis which is on both planes.

 Find a unit vector 𝐮u with positive first coordinate that is parallel to both planes.

 

(C) Use parts (A) and (B) to find a vector equation for the line of intersection of the two planes,𝐫(𝐭)=r(t)=

  

Answer 1

The Point P will have the form (0. q, 0) for some value of q.

It is easy to verify that q = 1/4 by observing that with q = 1/4 , the first equation becomes 1 = 1 and the second becomes 0 = 0

The unit vector can be found by working out the cross product between

(3,4,5) and ( 3,0,5) and normalizing.

The result is u = (5, 0 , -3)/sqrt(34)

The desired line of intersection is thus (0, 1/4 , 0 ) + t ( 5, 0, -4) / sqrt(34)