Projection of (2, 3) onto (-1, 1) Step by Step

Step by step

The question

Question: find the projection of (2, 3) onto (−1, 1).

v = [2; 3], s = [−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.

projection: the part of v along s

like the shadow of v on the line of s

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.

proj = v • ss • s × s

2 Work out the number

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

v • s = 2 × (−1) + 3 × 1 = −2 + 3 = 1

The bottom: s • s.

s • s = −1 × (−1) + 1 × 1 = 1 + 1 = 2

So the number is 1/2.

number = 12

Careful!

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

Not v dot v.

Careful: the bottom is s • s, not v • v

3 Times s

Times s: every entry times 1/2.

So the projection is (−1/2, 1/2).

proj = 12 × [−1; 1] = [−1 over 2; 1 over 2]

4 Draw it

The line through s.

s goes to (−1, 1).

v goes to (2, 3).

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

The projection ends where it lands: (−1/2, 1/2).

5 Check

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

v − proj = [2; 3] − [−1 over 2; 1 over 2]

v − proj = [2 − (−1 over 2); 3 − 1 over 2]

v − proj = [5 over 2; 5 over 2]

Its dot product with s is 0.

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

(v − proj) • s = 52 × (−1) + 52 × 1 = −52 + 52 = 0

Let's look back.

First: write the rule.

Then: work out the number.

Then: times s.

Then: draw it.

Last: check.

Answer