Are (2, -1, 1) and (1, 3, 1) orthogonal

Step by step

The question

Question: are (2, −1, 1) and (1, 3, 1) orthogonal?

Orthogonal means perpendicular: at a right angle.

Two vectors are orthogonal exactly when their dot product is 0.

1 Find the dot product

Matching entries multiplied, then added: 2 × 1 + (−1) × 3 + 1 × 1.

We work out each times.

Then we add: u • v = 0.

2 Compare with 0

The dot product is 0.

So yes: the vectors are orthogonal.

Careful!

Only a dot product of exactly 0 means orthogonal.

Any other number, even a small one, means not.

3 Check

Let's check with Pythagoras.

At a right angle, the length of u squared plus the length of v squared equals the length of u plus v, squared.

The length squared is the entries squared, added.

For u: 6.

For v: 11.

u plus v: (3, 2, 2).

Its length squared: 17.

6 plus 11 is 17, the same as 17.

Pythagoras works: a right angle.

So the answer is right!

Let's look back.

First: find the dot product.

Then: compare with 0.

Last: check.