Question

(a) Find parametric equations for the line through (3, 1, 8) that is perpendicular to the plane x − y + 4z = 7. (Use the parameter t.) (x(t), y(t), z(t)) = (b) In what points does this line intersect the coordinate planes?
xy-plane (x, y, z) =
yz-plane (x, y, z) =
xz-plane (x, y, z) =

Answer 1

Answer:

• (x, y, z) = (3+t, 1-t, 8+4t) . . . equation of the line

• xy-intercept (1, 3, 0)

• yz-intercept (0, 4, -4)

• xz-intercept (4, 0, 12)

Step-by-step explanation:

The line's direction vector is given by the coordinates of the plane: (1, -1, 4). So, the parametric equations can be ...

(x, y, z) = (3, 1, 8) + t(1, -1, 4) . . . . . parametric equation for the line

or

(x, y, z) = (3+t, 1-t, 8+4t)

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The various intercepts can be found by setting the respective variables to zero:

xy-plane: z=0, so t=-2. (x, y, z) = (1, 3, 0)

yz-plane: x=0, so t=-3. (x, y, z) = (0, 4, -4)

xz-plane: y=0, so t=1. (x, y, z) = (4, 0, 12)