Question

Where does the line through A(−3, 1, 0) and B(−1, 5, 6) intersect the plane 2x + y − z + 2 = 0? Justify all your steps

Answer 1

Explanation:

Answer 2

(0, 7, 9)

Explanation:

The direction vector (a, b, c) is given by B - A

(a, b, c) = (−1, 5, 6) - (-3, 1, 0) = (2, 4, 6)

(a, b, c) = (2, 4, 6)

Let us use point A as
`(x_o,y_o,z_o)`, therefore (-3, 1, 0) =
`(x_o,y_o,z_o)`

`x=x_o+at\\\\x=-3+2t\\\\y=y_o+bt\\\\y=1+4t\\\\z=z_o+ct\\\\z=0+6t`

Substituting the value of x, y and z into the plane equation:

2x + y − z + 2 = 0

2(-3+2t) + (1 + 4t) - (6t) + 2 = 0

-6 + 4t + 1 + 4t - 6t + 2 = 0

2t -3 = 0

2t = 3

`\x=-3+2t\\\\x=-3+2(3/2)=0\\\\y=1+4t\\\\y=1+4(3/2)=7\\\\z=0+6t\\\\z=6(3/2)=9\\\\x=0,y=7,z=9\\\\(0,7,9)`

t = 3/2