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Projection onto a vector

Linear Algebra · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Write the rule
  2. Work out the number
  3. Times s
  4. Draw it
  5. Check

The steps for Projection of (3, 4) onto (1, 0). Every solution names its steps, so you always know what comes next.

Worked examples

Projection of (3, 4) onto (1, 0) Easy

The question

Question: find the projection of (3, 4) onto (1, 0).

v = [3; 4], s = [1; 0]

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 × 1 + 4 × 0 = 3 + 0 = 3

The bottom: s • s.

s • s = 1 × 1 + 0 × 0 = 1 + 0 = 1

So the number is 3/1, which is 3.

number = 31 = 3

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 3.

So the projection is (3, 0).

proj = 3 × [1; 0] = [3; 0]

4 Draw it

The line through s.

s goes to (1, 0).

v goes to (3, 4).

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

The projection ends where it lands: (3, 0).

5 Check

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

v − proj = [3; 4] − [3; 0]
v − proj = [3 − 3; 4 − 0]
v − proj = [0; 4]

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 + 4 × 0 = 0 + 0 = 0

Let's look back.

First: write the rule.

Then: work out the number.

Then: times s.

Then: draw it.

Last: check.

Answer
proj = 3 × [1; 0] = [3; 0]

▶ Watch it solved

Projection of (1, 7) onto (1, 1) Medium

The question

Question: find the projection of (1, 7) onto (1, 1).

v = [1; 7], 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 = 1 × 1 + 7 × 1 = 1 + 7 = 8

The bottom: s • s.

s • s = 1 × 1 + 1 × 1 = 1 + 1 = 2

So the number is 8/2, which is 4.

number = 82 = 4

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 4.

So the projection is (4, 4).

proj = 4 × [1; 1] = [4; 4]

4 Draw it

The line through s.

s goes to (1, 1).

v goes to (1, 7).

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

The projection ends where it lands: (4, 4).

5 Check

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

v − proj = [1; 7] − [4; 4]
v − proj = [1 − 4; 7 − 4]
v − proj = [−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 = −3 × 1 + 3 × 1 = −3 + 3 = 0

Let's look back.

First: write the rule.

Then: work out the number.

Then: times s.

Then: draw it.

Last: check.

Answer
proj = 4 × [1; 1] = [4; 4]

▶ Watch it solved

Project (4, 1, 1) onto (1, 1, 0) Harder

The question

Question: find the projection of (4, 1, 1) onto (1, 1, 0).

v = [4; 1; 1], s = [1; 1; 0]

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 = 4 × 1 + 1 × 1 + 1 × 0 = 4 + 1 + 0 = 5

The bottom: s • s.

s • s = 1 × 1 + 1 × 1 + 0 × 0 = 1 + 1 + 0 = 2

So the number is 5/2.

number = 52

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 5/2.

So the projection is (5/2, 5/2, 0).

proj = 52 × [1; 1; 0] = [5 over 2; 5 over 2; 0]

4 Check

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

v − proj = [4; 1; 1] − [5 over 2; 5 over 2; 0]
v − proj = [4 − 5 over 2; 1 − 5 over 2; 1 − 0]
v − proj = [3 over 2; −3 over 2; 1]

Its dot product with s is 0.

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

(v − proj) • s = 32 × 1 + (−32) × 1 + 1 × 0 = 32 − 32 + 0 = 0

Let's look back.

First: write the rule.

Then: work out the number.

Then: times s.

Last: check.

Answer
proj = 52 × [1; 1; 0] = [5 over 2; 5 over 2; 0]

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. projection of (3, 4) onto (1, 0)
    Answer
    proj = 3 × [1; 0] = [3; 0]
  2. projection of (2, 5) onto (0, 1)
    Answer
    proj = 5 × [0; 1] = [0; 5]
  3. projection of (4, 2) onto (1, 1)
    Answer
    proj = 3 × [1; 1] = [3; 3]
  4. projection of (5, 1) onto (2, 0)
    Answer
    proj = 52 × [2; 0] = [5; 0]
  5. projection of (3, 1) onto (1, 2)
    Answer
    proj = 1 × [1; 2] = [1; 2]

Medium

  1. projection of (1, 7) onto (1, 1)
    Answer
    proj = 4 × [1; 1] = [4; 4]
  2. projection of (6, 2) onto (1, -1)
    Answer
    proj = 2 × [1; −1] = [2; −2]
  3. projection of (2, 4) onto (3, 1)
    Answer
    proj = 1 × [3; 1] = [3; 1]
  4. projection of (5, 0) onto (1, 2)
    Answer
    proj = 1 × [1; 2] = [1; 2]
  5. projection of (-1, 3) onto (1, 1)
    Answer
    proj = 1 × [1; 1] = [1; 1]
  6. projection of (1, 3) onto (2, 1)
    Answer
    proj = 1 × [2; 1] = [2; 1]
  7. project (1, 2, 3) onto (1, 1, 1)
    Answer
    proj = 2 × [1; 1; 1] = [2; 2; 2]
  8. project (3, 0, 3) onto (1, 2, 2)
    Answer
    proj = 1 × [1; 2; 2] = [1; 2; 2]

Harder

  1. project (4, 1, 1) onto (1, 1, 0)
    Answer
    proj = 52 × [1; 1; 0] = [5 over 2; 5 over 2; 0]
  2. project (2, 1, 3) onto (1, 0, 1)
    Answer
    proj = 52 × [1; 0; 1] = [5 over 2; 0; 5 over 2]
  3. projection of (1, 2) onto (3, 1)
    Answer
    proj = 12 × [3; 1] = [3 over 2; 1 over 2]
  4. project (1, -1, 5) onto (1, 2, 2)
    Answer
    proj = 1 × [1; 2; 2] = [1; 2; 2]
  5. projection of (2, 3) onto (-1, 1)
    Answer
    proj = 12 × [−1; 1] = [−1 over 2; 1 over 2]
  6. projection of (3, -2) onto (2, 3)
    Answer
    proj = 0 × [2; 3] = [0; 0]
  7. project (2, 2, 1, 1) onto (1, 1, 1, 1)
    Answer
    proj = 32 × [1; 1; 1; 1] = [3 over 2; 3 over 2; 3 over 2; 3 over 2]