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

Step by step

The question

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

v = [3; −2], s = [2; 3]

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 = 3 × 2 + (−2) × 3 = 6 − 6 = 0

The bottom: s • s.

s • s = 2 × 2 + 3 × 3 = 4 + 9 = 13

So the number is 0/13, which is 0.

number = 013 = 0

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 0.

So the projection is (0, 0).

proj = 0 × [2; 3] = [0; 0]

4 Draw it

The line through s.

s goes to (2, 3).

v goes to (3, −2).

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

The projection ends where it lands: (0, 0).

5 Check

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

v − proj = [3; −2] − [0; 0]

v − proj = [3 − 0; −2 − 0]

v − proj = [3; −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 = 3 × 2 + (−2) × 3 = 6 − 6 = 0

Let's look back.

First: write the rule.

Then: work out the number.

Then: times s.

Then: draw it.

Last: check.

Answer