Equation of a Plane in Three Dimensions
A plane \(\Pi\) passes through the points A(2,2,-1), B(3,2,-1), C(4,3,5). Find the equation of the plane \(\Pi\) in the form \(\mathbf{a}+\lambda \mathbf{b}+\mu \mathbf{c}\)
- Use \(\mathbf{a}\) as starting point: \(a \to b=(1,0,0)\), \(a \to c=2,1,6\)
\(\(\mathbf{r}=\begin{pmatrix}2 \\2 \\-1\end{pmatrix}+\lambda \begin{pmatrix}1 \\0 \\0\end{pmatrix}+\mu \begin{pmatrix}2 \\1 \\6\end{pmatrix}\)\)
Verify that the point \(\begin{pmatrix}2 \\ 2 \\ -1\end{pmatrix}\) lies in the plane with equation, \(\mathbf{r}=\begin{pmatrix}3 \\ 4 \\ -2\end{pmatrix}+\lambda \begin{pmatrix}2 \\ 1 \\ 1\end{pmatrix}+\mu \begin{pmatrix}1 \\ -1 \\ 2\end{pmatrix}\)
- \(\begin{pmatrix}2 \\ 2 \\ 1\end{pmatrix}=\begin{pmatrix}3 \\ 4 \\ -2\end{pmatrix}+\lambda \begin{pmatrix}2 \\ 1 \\ 1\end{pmatrix}+\mu \begin{pmatrix}1 \\ -1 \\ 2\end{pmatrix}\)
- \(\begin{pmatrix}2 \\ 2 \\ -1\end{pmatrix}=\begin{pmatrix}3+2\lambda+\mu \\4+\lambda+\mu \\-2+\lambda+2\mu\end{pmatrix}\)
- Use one of these:
\(2=3+2\lambda+\mu\)
\(2\lambda+\mu+1=0\)
\(2=4+\lambda-\mu\)
\(\lambda-\mu+2=0\)
\(\lambda=-1, \mu=1\)
The points A B and C have vector equations \(\begin{pmatrix}1 \\ 3 \\ 2\end{pmatrix},\begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix},\begin{pmatrix}2 \\ 1 \\ 0\end{pmatrix}\). The plane \(\Pi\) contains points A B and C. Find an equation for \(\Pi\).
- \(A \to B=\begin{pmatrix}-2 \\ -3 \\ -1\end{pmatrix}\)
- \(A \to C=\begin{pmatrix}1 \\ -2 \\ -2\end{pmatrix}\)
- Therefore our equation is:
\(\begin{pmatrix}1 \\ 3 \\ 2\end{pmatrix}+\lambda \begin{pmatrix}-2 \\ -3 \\ -1\end{pmatrix}+\mu \begin{pmatrix}1 \\ -2 \\ -2\end{pmatrix}\)
The plane \(\Pi\) is perpendicular to the normal \(\mathbf{n}=3\mathbf{i}-2\mathbf{j}+\mathbf{k}\) and passes through the point P with position vector \(8\mathbf{i}+4\mathbf{j}-7\mathbf{k}\).
\(3x-2y+z=c\)
\(3(8)-2(4)-7=c\)
\(24-8-7=c\)
\(c=9\)
\(3x-2y+z=9\)
The line \(l_{1}\) is normal to the plane \(\Pi\) with Cartesian equation \(5x- 3y - 4z = 9\) and passes through the point \((2, 3, —2)\). Find: a) a vector equation of \(l_{1}\), b) a Cartesian equation of \(l_{1}\).
a) \(\begin{pmatrix}2 \\ 3 \\ -2\end{pmatrix}+\lambda\begin{pmatrix}5 \\ -3 \\ -4\end{pmatrix}\)
b) \(\frac{x-2}{5}+\frac{y-3}{-3}+\frac{z+2}{-4}\)