Plane through (4, -1, 2) with normal <-3, 1, 5>

Step by step

The question

Question: find the plane through (4, −1, 2) with normal vector n = ⟨−3, 1, 5⟩.

1 Draw it

A normal vector is an arrow at a square corner to the plane.

The plane goes through the point,

and the normal sticks straight out of it.

2 Write the plane

The rule: a(x − x₀) + b(y − y₀) + c(z − z₀) = 0.

The normal gives a, b, c.

The point gives x₀, y₀, z₀.

Careful!

The normal's numbers go in front.

The point's numbers go inside the brackets, with a minus.

3 Open the brackets

What is in front of a bracket multiplies every part inside.

A minus in front changes the signs inside.

Add the plain numbers: they make 3.

Move the number to the right: take away 3 on both sides.

So the plane is −3x + y + 5z = −3.

4 Check

Let's check: put the point (4, −1, 2) in.

We get −3.

Right!

Let's look back.

First: draw it.

Then: write the plane.

Then: open the brackets.

Last: check.