Question

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Calc III question about making two vectors from a given plane.(3 points in 3-d) @iambatman

Answer 1

so if youre given a question such as find a vector perpendicular to the plane that passes through the points, A,B,C. where each point has a (x,y,z) coordinate

Answer 2

to get a vector perpendicular you need to do the cross product, i understand that to use the cross product both vectors need to start from the same point, but does it matter which one they start from. as long as it is consistent?

Answer 3

@iambatman

Answer 4

@SithsAndGiggles

Answer 5

You have to find the cross product of any two vectors in the plane. For example, say you have the plane
\[2x+y-z=0\]
Pick any two vectors \(\langle x,y,z\rangle\) that satisfy this equation (other than the zero vector in this case). Of course the vectors also can't be (anti)parallel either.

Say I pick \(v_1=\langle 1,1,3\rangle\) and \(v_2=\langle0,1,1\rangle\). Then a vector normal/orthogonal to both of these is
\[\begin{align*}
v_1\times v_2&=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\
1&1&3\\
0&1&1\end{vmatrix}\\\\
&=\begin{vmatrix}1&3\\1&1\end{vmatrix}\vec{i}-\begin{vmatrix}1&3\\0&1\end{vmatrix}\vec{j}+\begin{vmatrix}1&1\\0&1\end{vmatrix}\vec{k}\\\\
&=\langle-2,-1,1\rangle
\end{align*}\]
Note that \(\langle2,1,-1\rangle\) is also normal to these vectors; it just faces the opposite direction.

Answer 6

im asking if you are given 3 points not vectors

Answer 7

Ah sorry. You'll find the info you need here. Plenty of examples.

http://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx

Answer 8

shweet thanks