Determine an equation of the plane containing the following points: (2,0,2), (4,3,1), and (0,1,3).
Do I set up a cross-product?
Determine an equation of the plane containing the following points: (2,0,2), (4,3,1), and (0,1,3).
Do I set up a cross-product?
Hello, riverjib!
Yes, we need a cross-product . . . but are you just guessing?Determine an equation of the plane containing the points: .$\displaystyle A(2,0,2),\;B(4,3,1),\;C(0,1,3)$
. . Do you have any idea why?
To write the equation of a plane, we need a point on the plane, $\displaystyle (x_1,y_1,z_1)$
. . (we have 3 to choose from), and the normal vector of the plane, $\displaystyle \vec n \,=\,\langle a,b,c\rangle$.
Then substitute into: .$\displaystyle a(x-x_1) + b(y - y_1) + c(z-z_1) \:=\:0$
We have: .$\displaystyle AB \:=\:\langle2,3,-1\rangle,\;\;AC \:=\:\langle -2,1,1\rangle $
. . Then: .$\displaystyle \vec n \;=\;\begin{vmatrix}i & j & k \\ 2 & 3 & \text{-}1 \\ \text{-}2 & 1 & 1 \end{vmatrix} \;=\;i(3+1) -j(2-2) + k(2+6) \;=\;4i + 8k$
Hence: .$\displaystyle \vec n \;=\;\langle4,0,8\rangle \;=\;\langle1,0,2\rangle$
Can you finish the problem now?