Multivariable calculus and vector analysis

Forming plane examples

Example 1

Find the equation for the plane through the point $\left(0,1,-7\right)$ perpendicular to the vector $\left(4,-1,6\right)$.

Solution: Let $\mathbf{a}=\left(0,1,-7\right)$. Let $\mathbf{n}=\left(4,-1,6\right)$. Then, for $\mathbf{x}=\left(x,y,z\right)$, the equation for the plane is

$$\begin{array}{lll}\hfill \mathbf{n}\cdot \left(\mathbf{x}-\mathbf{a}\right)=0.& & \hfill \end{array}$$

This becomes

$$\begin{array}{lll}\hfill \left(4,-1,6\right)\cdot \left(x-0,y-1,z+7\right)=0& & \hfill \end{array}$$

or

$$\begin{array}{lll}\hfill 4x-\left(y-1\right)+6\left(z+7\right)=0.& & \hfill \end{array}$$

Often, we prefer to write this as

$$\begin{array}{lll}\hfill 4x-y+6z+43=0.& & \hfill \end{array}$$

(Note that you can read the normal vector $\mathbf{n}=\left(4,-1,6\right)$ right from the equation for the plane; the components of $\mathbf{n}$ are simply the coefficients of $x$, $y$, and $z$.)

Example 2

Find the equation for the plane through the points $\left(0,1,-7\right)$, $\left(3,1,-9\right)$, and $\left(0,-5,-8\right)$.

Solution: Let $\mathbf{b}=\left(0,1,-7\right)-\left(3,1,-9\right)=\left(-3,0,2\right)$. Let $\mathbf{c}=\left(0,1,-7\right)-\left(0-5,-8\right)=\left(0,6,1\right)$. Then, a normal vector is

$$\begin{array}{llll}\hfill \mathbf{n}& =\mathbf{b}\times \mathbf{c}& \hfill & \\ \hfill & =\left|\begin{array}{ccc}\hfill \mathbf{i}& \hfill \mathbf{j}& \hfill \mathbf{k}\\ \hfill -3& \hfill 0& \hfill 2\\ \hfill 0& \hfill 6& \hfill 1\end{array}\right|& \hfill & \\ \hfill & =\mathbf{i}\left(0-12\right)-\mathbf{j}\left(-3-0\right)+\mathbf{k}\left(-18-0\right)& \hfill & \\ \hfill & =\left(-12,3,-18\right).& \hfill & \end{array}$$

We’ll pick the first point and let $\mathbf{a}=\left(0,1,-7\right)$. The equation for the plane becomes

$$\begin{array}{lll}\hfill \left(-12,3,-18\right)\cdot \left(x-0,y-1,z+7\right)=0& & \hfill \end{array}$$

which we rewrite as

$$\begin{array}{lll}\hfill -12x+3\left(y-1\right)-18\left(z+7\right)=0& & \hfill \end{array}$$

or

$$\begin{array}{lll}\hfill -12x+3y-18z-129=0.& & \hfill \end{array}$$

I hope you noticed that the planes from both examples when through the same point $\left(0,1,-7\right)$. Did you also notice that their normal vectors are parallel to each other? If you multiply the $\mathbf{n}$ from Example 1 by $-3$, you obtain the $\mathbf{n}$ from Example 2:

$$\begin{array}{lll}\hfill -3\left(4,-1,6\right)=\left(-12,3,-18\right).& & \hfill \end{array}$$

What does that mean about the relationship between the two planes? (This was discussed in the pre-lecture reading.) The two planes must be equal. In fact, if you divide both sides of the equation of the second plane by $-3$, you get the equation of the first plane.

Example 3

Find the equation for the plane through the points $\left(1,2,3\right)$, $\left(2,4,6\right)$, and $\left(-3,-6,-9\right)$.

Solution: Let $\mathbf{b}=\left(2,4,6\right)-\left(1,2,3\right)=\left(1,2,3\right)$ and let $\mathbf{c}=\left(1,2,3\right)-\left(-3,-6,-9\right)=\left(4,8,12\right)$. Then, we attempt to find a normal vector as their cross product:

$$\begin{array}{llll}\hfill \mathbf{n}& =\mathbf{b}\times \mathbf{c}& \hfill & \\ \hfill & =\left|\begin{array}{ccc}\hfill \mathbf{i}& \hfill \mathbf{j}& \hfill \mathbf{k}\\ \hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 8& \hfill 12\end{array}\right|& \hfill & \\ \hfill & =\mathbf{i}\left(24-24\right)-\mathbf{j}\left(12-12\right)+\mathbf{k}\left(8-8\right)& \hfill & \\ \hfill & =\left(0,0,0\right).& \hfill & \end{array}$$

Hmm, something went wrong, as we weren’t able to find a normal direction. Do the three points actually determine a plane? If you are still puzzled, go back to the pre-lecture reading, and look for any conditions on the ability of three points to determine a plane.

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

1b. Lines and planes

• Parametrization of a line
• Forming planes
• A line or a plane or a point?
• Parametrization of a plane
• Distance from point to plane
• Parametrization of a line examples
• Forming plane examples
• Plane parametrization example
• Distance from point to plane example
• Intersecting planes example

1c. Matrices

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material