# How do you check if 2 vectors are orthogonal?

1. Nov 21, 2012

### charlies1902

How do you check if 2 vectors are orthogonal?

I know that if 2 vectors are orthogonal, then there dot product is 0. But I don't think that necessarily means if their dot product is 0, the 2 vectors are orthogonal. Like what if you had 2 zero vectors, their dot produt would be 0, but they're not orthogonal.

I also know that the angle between the 2 vectors is 90 degrees. I think this one that 2 vectors that are 90 degrees apart are orthogonal. Right?

2. Nov 21, 2012

### Staff: Mentor

Sure they are. The zero vector is considered to be orthogonal to every other vector, including another zero vector.
Right.

3. Nov 21, 2012

### charlies1902

Okay thank you.

Also if we have 2 vectors u and v, and vector w is the projection of u onto v.
How is the length of w determined?

4. Nov 21, 2012

### haruspex

It depends on the projection. If you mean an orthogonal projection (i.e. orthogonal to v) then it will satisfy (u-w).v = 0

5. Nov 21, 2012

### charlies1902

Uh, for this problem I'm doing the instructions are "The vector w is called the orthogonal projection of u
onto v. Sketch the three vectors u, v, and w."

I attached my work.
Vector u is given to be [-2 3]
v is given to be [4 0]

I calculated the orthogonal projection of u onto v
by 1st finding the dot product of u and v.
Then dividing that by the magnitude of vector v squared.
Then multipling that by vector v.
This gave:
w=[-2 0]

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6. Nov 21, 2012

### haruspex

Looks like a valid method and the right answer. And it does satisfy (u-w).v = 0.

7. Nov 21, 2012

### charlies1902

I think for this one you can find out by inspection since vector v is on the x axis. I jsut saw that the projection equation I used is introduced in a later section, so I probably should have used another method.
What other method is there if this were not an obvious case where v is not on the x axis? As in how do you find the length of vector w.

8. Nov 21, 2012

### haruspex

I don't see an easier way than deriving that equation.
It's clear that w = λv for some scalar λ. And orthogonality gives (u-w).v = 0, right?
Substitute for w and determine λ.