Answer:
(x(t), y(t), z(t)) = (4 + t, 1 - t, 8 + 4t)
xy - plane (x, y, z) = (2, -1, 0)
yz - plane (x, y, z) = (0, 5, -8)
xz - plane (x, y, z) = (5, 0, 12)
Step-by-step explanation:
The given point (x, y ,z) = (4, 1, 8)
The plane x -y + 4z = 2
Normal vector (n) = < 1, -1, 4 >
The equation of line through point (4, 1, 8) and the plane is:
(x(t), y(t), z(t)) = (4, 1, 8) + t(1, -1, 4)
(x(t), y(t), z(t)) = (4 + t, 1 - t, 8 + 4t)
Any point on the line P(x, y, z) = ( 4 + t, 1 - t, 8 + 4t)
xy-Pane ⇒ z = 0
8 + 4t = 0
4t = - 8
t = -8/4
t = -2
∴
(x, y, z) = (4 - 2, 1 - 2, 8 + 4(-2))
(x, y, z) = (2, -1, 0)
yz-plane ⇒ x = 0
4 + t = 0
t = -4
∴
(x, y, z) = (4 + (-4) , 1-(-4), 8 + 4(-4)
(x, y, z) = (0, 5, -8)
xz-plane ⇒ y = 0
1 - t = 0
-t = -1
t = 1
∴
(x, y, z) = ( 4 + 1, 1 - 1, 8 + 4(1) )
(x, y, z) = (5, 0, 12)
