Determine parametric equations for the plane through the points A(2, 1, 1), B(0, 1, 3), and C(1, 3, −2).
If the plane has equation ax+by+cd = m then we need
2a+b+c = m
b+3c = m
a+3b-2c = m
If I have done my math right,
a = m/5, b=2m/5, c=m/5
So, if we set m=5 to clear the fractions, we get
x+2y+z = 5
Now, z = 5-x-2y
So, if we let
x = s
y = t
z = 5-s-2t
To find the parametric equations for the plane through the points A(2, 1, 1), B(0, 1, 3), and C(1, 3, −2), we can use the fact that a plane is determined by a point on the plane and two non-collinear direction vectors.
1. Find a direction vector:
To find a direction vector, we can take the two vectors formed by the points A, B, and C, and calculate their cross product. The cross product of two vectors will give us a vector that is orthogonal (perpendicular) to both of them.
Let's take the vectors formed by AB and AC:
AB = B - A = (0 - 2, 1 - 1, 3 - 1) = (-2, 0, 2)
AC = C - A = (1 - 2, 3 - 1, -2 - 1) = (-1, 2, -3)
Now, let's calculate their cross product:
n = AB x AC = (-2, 0, 2) x (-1, 2, -3)
The cross product can be calculated as:
n = (0 - 2(2), 2(-2) - (-3)(-1), (-2)(2) - 0) = (-4, -1, -4)
So, a direction vector of the plane is n = (-4, -1, -4).
2. Choose a point on the plane:
We can choose any of the given points on the plane as a reference point. Let's choose point A(2, 1, 1).
3. Write the parametric equations:
The parametric equations for the plane can be written as:
x = A.x + m * AB.x + n * AC.x
y = A.y + m * AB.y + n * AC.y
z = A.z + m * AB.z + n * AC.z
where m and n are parameters that can take any real value.
Substituting the values:
x = 2 + m * (-2) + n * (-1)
y = 1 + m * 0 + n * 2
z = 1 + m * 2 + n * (-3)
Therefore, the parametric equations for the plane through the points A(2, 1, 1), B(0, 1, 3), and C(1, 3, −2) are:
x = 2 - 2m - n
y = 1 + 2n
z = 1 + 2m - 3n
That's it! We have determined the parametric equations for the given plane.