How do you check if 2 vectors are orthogonal?

[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?

 

[Mark44]

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.

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

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?

Right.

 

[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?

 

[haruspex]

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?

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

 

[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]

 

[haruspex]

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]

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

 

[charlies1902]

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

I think for this one you can find out by inspection since vector v is on the x axis. I just 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.

 

[haruspex]

I think for this one you can find out by inspection since vector v is on the x axis. I just 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.

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.