Project (1, -1, 5) onto (1, 2, 2) Step by Step

Step by step

The question

Question: find the projection of (1, −1, 5) onto (1, 2, 2).

v = [1; −1; 5], s = [1; 2; 2]

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 = 1 × 1 + (−1) × 2 + 5 × 2 = 1 − 2 + 10 = 9

The bottom: s • s.

s • s = 1 × 1 + 2 × 2 + 2 × 2 = 1 + 4 + 4 = 9

So the number is 9/9, which is 1.

number = 99 = 1

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.

So the projection is (1, 2, 2).

proj = 1 × [1; 2; 2] = [1; 2; 2]

4 Check

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

v − proj = [1; −1; 5] − [1; 2; 2]

v − proj = [1 − 1; −1 − 2; 5 − 2]

v − proj = [0; −3; 3]

Its dot product with s is 0.

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

(v − proj) • s = 0 × 1 + (−3) × 2 + 3 × 2 = 0 − 6 + 6 = 0

Let's look back.

First: write the rule.

Then: work out the number.

Then: times s.

Last: check.

Answer