What is the dot product of the vectors <3, -1> and <2, 4>? Step by Step

Step by step

The question

Question: find the dot product of u = ⟨3, −1⟩ and v = ⟨2, 4⟩.

u = ⟨3, −1⟩ and v = ⟨2, 4⟩

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Multiply the matching parts

The dot product: across times across, plus up times up.

u • v = 3 × 2 + −1 × 4

3 times 2 is 6.

−1 times 4 is −4.

u • v = 6 − 4

6 minus 4 is 2.

So u • v = 2.

u • v = 2

Careful!

The dot product is not a vector.

We add the two products.

The answer is one number.

Careful: not ⟨6, −4⟩ ✗

2 Check

Does it make sense?

Let's draw u and v.

The angle between them is less than a square corner, 90°.

Why does the sign tell us the angle?

The next lesson shows: u • v is the two lengths times cos θ.

Lengths are never negative.

So u • v has the sign of cos θ: plus below 90°, minus above, 0 at 90°.

u • v = ‖u‖ × ‖v‖ × cos θ

A positive dot product means the angle is less than 90°.

2 is positive.

It fits!

2 > 0 → less than 90°

3 Pick the choice

Our answer is 2.

That is choice A.

A) 2

4 Why not the others

B is 6.

Wrong: only the first numbers: we add both products.

B) 6 ✗ only the first numbers: we add both products

C is 8.

Wrong: added the matching numbers: we multiply them.

C) 8 ✗ added the matching numbers: we multiply them

D is 10.

Wrong: (−1) × 4 is −4, not +4.

D) 10 ✗ (−1) × 4 is −4, not +4

Let's look back.

First: multiply the matching parts.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer