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

Step by step

The question

Question: are (1, 1, 1, 1) and (1, −1, 1, −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: 1 × 1 + 1 × (−1) + 1 × 1 + 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: 4.

For v: 4.

u plus v: (2, 0, 2, 0).

Its length squared: 8.

4 plus 4 is 8, the same as 8.

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.