The Cross Product and Planes

Section2.5The Cross Product and Planes

ObjectivesTopical

use the normal vector to find the equation for a plane

We will now combine the dot product with the cross product to develop an equation of a plane in 3D. Before doing so, let's look at what information we need to obtain a line in 2D, and a plane in 3D. To obtain a line in 2D, one way is to have 2 points. The next problem introduces the new idea by showing you how to find an equation of a line in 2D.

Subsection2.5.1Planes from Points

Exercise2.5.1

Suppose the point \(P=(1,2)\) lies on line \(L\text{.}\) Suppose that the angle between the line and the vector \(\vec n=\left\lt 3,4\right>\) is 90\(^\circ\) (whenever this happens we say the vector \(\vec n\) is normal to the line). Let \(Q=(x,y)\) be another point on the line \(L\text{.}\) Use the fact that \(\vec n\) is orthogonal to \(\vec {PQ}\) to obtain an equation of the line \(L\text{.}\)

Exercise2.5.2

Let \(P=(a,b,c)\) be a point on a plane in 3D. Let \(\vec n=\langle A,B,C \rangle\) be a normal vector to the plane (so the angle between the plane and and \(\vec n\) is 90\(^\circ\)). Let \(Q=(x,y,z)\) be another point on the plane. Show that an equation of the plane through point \(P\) with normal vector \(\vec n\) is

\begin{equation*} A(x-a)+B(y-b)+C(z-c)=0. \end{equation*}

Exercise2.5.3

Consider the three points \(P=(1,0,0), Q=(2,0,-1), R=(0,1,3)\text{.}\) Find an equation of the plane which passes through these three points.

Hint

First find a normal vector to the plane.

Subsection2.5.2Planes and Lines

Exercise2.5.4

Find an equation of the plane containing the lines \(\vec r_1(t)=(1,3,0)t+(1,0,2)\) and \(\vec r_2(t)=(2,0,-1)t+(2,3,2)\text{.}\)

Exercise2.5.5

Consider the points \(P=(2,-1,0)\text{,}\) \(Q=(0,2,3)\text{,}\) and \(R=(-1,2,-4)\text{.}\)

(a)

Give an equation \((x,y,z)=(?,?,?)\) of the line through \(P\) and \(Q\text{.}\)

(b)

Give an equation of the line through \(P\) and \(R\text{.}\)

(c)

Give an equation of the plane through \(P\text{,}\) \(Q\text{,}\) and \(R\text{.}\)

Exercise2.5.6

Consider the two planes \(x+2y+3z=4\) and \(2x-y+z=0\text{.}\) These planes meet in a line. Find a vector that is parallel to this line. Then find a vector equation of the line.

Subsection2.5.3Computational Practice

These are provided to help you achieve better skills in basic computational answers