A plane contains the point B(-3, 2, -4) and the line with parametric equations x = 1 + 2t, y = -t, z = -2 + 3t. Determine vector and parametric equations of the plane.
To determine the vector equation of a plane, we need a point on the plane and a normal vector to the plane.
Given a point B(-3, 2, -4) on the plane, we can use it as our reference point.
To find the normal vector, we need two non-parallel vectors that lie on the plane.
From the given line with parametric equations:
x = 1 + 2t
y = -t
z = -2 + 3t
We can choose any two values of t to find two corresponding points on the line.
Let's use t = 0 and t = 1 to find two points, P1 and P2.
When t = 0:
x = 1 + 2(0) = 1
y = -(0) = 0
z = -2 + 3(0) = -2
When t = 1:
x = 1 + 2(1) = 3
y = -(1) = -1
z = -2 + 3(1) = 1
So, we have:
P1(1, 0, -2)
P2(3, -1, 1)
We can now find two vectors on the plane using these points:
Vector V1 = P1B = (-3 - 1, 2 - 0, -4 - (-2)) = (-4, 2, -2)
Vector V2 = P1P2 = (3 - 1, -1 - 0, 1 - (-2)) = (2, -1, 3)
Now, we can find the normal vector N to the plane by taking the cross product of V1 and V2:
N = V1 x V2 = (2 * (-2) - (-1) * (-2), (-4) * 3 - 2 * (-2), (-4) * (-1) - 2 * 2)
= (-2, -8, 0)
Therefore, the normal vector to the plane is N = (-2, -8, 0).
Finally, we can write the vector equation of the plane by using the point B(-3, 2, -4) and the normal vector N as follows:
<plane equation>
(r - B) · N = 0
Substituting the values we have:
(r - (-3, 2, -4)) · (-2, -8, 0) = 0
Simplifying further:
(-3 - x, 2 - y, -4 - z) · (-2, -8, 0) = 0
(-3 - x)(-2) + (2 - y)(-8) + (-4 - z)(0) = 0
6 + 2x -16 + 8y - 4z = 0
2x + 8y - 4z - 10 = 0
So, the vector equation of the plane is:
2x + 8y - 4z - 10 = 0
To find the parametric equations of the plane, we can express x, y, and z in terms of a parameter s:
2x + 8y - 4z - 10 = 0
Solving for x:
2x = -8y + 4z + 10
x = -4y + 2z + 5
Therefore, the parametric equations of the plane are:
x = -4y + 2z + 5
y = y (where y can take any value)
z = z (where z can take any value)
So, the parametric equations are:
x = -4y + 2z + 5
y = y
z = z