Question

A line ℓ passes through the point P(1,4,2) and is parallel to the vector ⟨3,2,−2⟩. At what point Q does ℓ intersect the xy plane?

1. Q(0,4,2)
2. Q(0,2,−2)
3. Q(−2,2,0)
4. Q(0,−2,6)
5. Q(4,6,0)
6. Q(6,4,0)

Answer 1

The point of intersection with the xy-plane is Q(4,6,0).

Step-by-step explanation:

To find the point of intersection with the xy-plane, we can set the z-coordinate of the point Q to 0 since it lies in the xy-plane. We know that the line is parallel to the vector [3,2,-2] and passes through the point P(1,4,2). So, we can write the equation of the line as follows:

x = 1 + 3t

y = 4 + 2t

z = 2 - 2t

Substituting z = 0 into the equation, we get:

2 - 2t = 0

2t = 2

t = 1

Now, substitute t = 1 into the equations for x and y:

x = 1 + 3(1) = 4

y = 4 + 2(1) = 6

Therefore, the point Q of intersection with the xy-plane is Q(4,6,0).