13.ReviewEx47-56 - Chapter Review Exercises 443 We want to...

Chapter Review Exercises 443
We want to find the value of t for which the point ( x , y , z ) lies on the plane. We substitute the parametric equations in
the equation of the plane and solve for t :
2 ( 3 t + 2 ) − 3 · 1 + ( − 7 t ) = 5
6 t + 4 − 3 − 7 t = 5
− t = 4 ⇒ t = − 4
The point P of intersection of the line and the plane has the coordinates:
x = 3 · ( − 4 ) + 2 = − 10 , y = 1 , z = − 7 · ( − 4 ) = 28
and thus,
P = ( − 10 , 1 , 28 ) .
47. Find the trace of the plane 3 x − 2 y + 5 z = 4 in the xy -plane.
SOLUTION
The xy -plane has equation z = 0, therefore the intersection of the plane 3 x − 2 y + 5 z = 4 with the
xy -plane must satisfy both z = 0 and the equation of the plane. Therefore the trace has the following equation:
3 x − 2 y + 5 · 0 = 4 ⇒ 3 x − 2 y = 4
We conclude that the trace of the plane in the xy -plane is the line 3 x − 2 y = 4 in the xy -plane.
48. Find the intersection of the planes x + y + z = 1 and 3 x − 2 y + z = 5.
SOLUTION
The line of intersection of the planes x + y + z = 1 and 3 x − 2 y + z = 5 consists of all points that satisfy
the equations of the two planes. Therefore, we must solve the following equations:
x + y + z = 1
3 x − 2 y + z = 5
The first equation implies y = 1 − x − z . Substituting in the second equation and solving for x in terms of z