Projection of (6, 2) onto (1, -1)

Step by step

The question

Question: find the projection of (6, 2) onto (1, −1).

The projection of v onto s is the part of v that goes along s.

Think of the shadow of v on the line through s, with the light straight above.

1 Write the rule

The rule: v dot s, over s dot s, times s.

The fraction is one number: how much of s we need.

2 Work out the number

The top: v • s, matching entries multiplied and added.

The bottom: s • s.

So the number is 4/2, which is 2.

Careful!

The bottom is s dot s, the vector we project onto.

Not v dot v.

3 Times s

Times s: every entry times 2.

So the projection is (2, −2).

4 Draw it

The line through s.

s goes to (1, −1).

v goes to (6, 2).

From the tip of v, straight down onto the line: at a right angle.

The projection ends where it lands: (2, −2).

5 Check

Let's check: the leftover, v minus the projection, must be at a right angle to s.

Its dot product with s is 0.

So the leftover is at a right angle to s: the answer is right!

Let's look back.

First: write the rule.

Then: work out the number.

Then: times s.

Then: draw it.

Last: check.