Write down an equation for the plane passing through the points [1,−2,−1][1,−2,−1], [−2,3,−3][−2,3,−3], and [2,0,3][2,0,3]. For example: x+y+z=1.
Question
Write down an equation for the plane passing through the points
- P1:
- P2:
- P3:
For example: .
Solution
Sure, here are the steps to find the equation of the plane passing through the points (1,-2,-1), (-2,3,-3), and (2,0,3).
Step 1: Find two vectors that are in the plane. Vector AB = B - A = (-2-1, 3-(-2), -3-(-1)) = (-3, 5, -2) Vector AC = C - A = (2-1, 0-(-2), 3-(-1)) = (1, 2, 4)
Step 2: Find the normal to the plane by taking the cross product of AB and AC. N = AB x AC = (5*-4 - -22, -21 - -34, -32 - 1*5) = (-20+4, -2+12, -6-5) = (-16, 10, -11)
Step 3: Use the normal and a point on the plane to find the equation of the plane. The equation of a plane is given by N . (X - A) = 0, where N is the normal, X is a point on the plane, and A is a known point on the plane. Substituting the values we have:
-16(x - 1) + 10(y - -2) - 11(z - -1) = 0 -16x + 16 + 10y + 20 - 11z - 11 = 0 -16x + 10y - 11z + 25 = 0
So, the equation of the plane is -16x + 10y - 11z + 25 = 0.
Similar Questions
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