Consider the planes 4x15x22x328 3x15x24x314 Find a point P t

Consider the planes

4x1+5x2+2x3=28

3x15x2+4x3=14

Find a point P that is on both planes. P:

Find a vector v that is parallel to both planes. v=

Find a vector equation for the intersection of the two planes in the form x(t)=.(............) + t(.........)

4x1+5x2+2x3=28

3x15x2+4x3=14

fix x3 =0

4x1+5x2=28

3x15x2=14

add

4x1+5x2 +3x15x2=-28+14

7x1=-14

x1=-2

4x1+5x2=28

4(-2)+5x2=28

5x2=20

x2=4

point P that is on both planes. P:(x1,x2,x3)=(-2,-4,0)

normal of 4x1+5x2+2x3=28 is n₁=<4,5,2>

normal of 3x15x2+4x3=14 is n₂=<3,-5,4>

vector v that is parallel to both planes. v=n₁ x n₂

v=<(5*4) -((-5)*2) ,(3*2)-(4*4) ,(4*(-5))-(3*5)>

v=<30 ,-10 ,-35>

vector v that is parallel to both planes. v=<30 ,-10 ,-35>

vector equation for the intersection of the two planes in the form r(t)=.(-2,-4,0)+ t<30 ,-10 ,-35>