A plane passes through A(1, 2, 3), B(1, -1, 0) and C(2, -3, -4). Determine vector and parametric equations of the plane.
To determine the vector equation of the plane, we need to find two vectors that lie on the plane. Let's take vectors AB and AC.
Vector AB = B - A = (1, -1, 0) - (1, 2, 3) = (0, -3, -3)
Vector AC = C - A = (2, -3, -4) - (1, 2, 3) = (1, -5, -7)
Now, we will use the point A(1, 2, 3) as a reference point on the plane.
The vector equation of the plane can be written as:
r = A + s * AB + t * AC
where r is the position vector (x, y, z) on the plane, s and t are scalar multiples.
Substituting the values we found:
r = (1, 2, 3) + s * (0, -3, -3) + t * (1, -5, -7)
Simplifying, we get the vector equation of the plane:
r = (1 + s, 2 - 3s - 5t, 3 - 3s - 7t)
To obtain the parametric equations of the plane, we can express x, y, and z in terms of parameters.
x = 1 + s
y = 2 - 3s - 5t
z = 3 - 3s - 7t
These are the parametric equations representing the plane passing through points A, B, and C.