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13.ReviewEx47-56

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

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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
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