Unit Vector Geometry: Find u.v & u.w

[knowLittle]

Homework Statement

If u is a unit vector, find u.v and u.w

Homework Equations

I assumed that unit vector means u=<1,1,1>
u.v=|u||v|cos60
My knowledge of unit vectors is very limited. I know that a unit vector is
i=<1,0,0>
j=<0,1,0>
k=<0,0,1>

The Attempt at a Solution

Since the triangle is isoceles. |u|=|v|
u.v=(sqrt(3)+sqrt(3) )cos60=-3.29925

Is this correct?

 

[knowLittle]

An image of the above problem has been attached.

 

[knowLittle]

Can someone answer?

 

[Mark44]

Homework Statement

If u is a unit vector, find u.v and u.w

Homework Equations

I assumed that unit vector means u=<1,1,1>

No, this is not a unit vector. It's not a unit vector because its length is not 1. (Its length is √3.)

u.v=|u||v|cos60
My knowledge of unit vectors is very limited. I know that a unit vector is
i=<1,0,0>
j=<0,1,0>
k=<0,0,1>

Yes, these are three unit vectors, but a unit vector can point in any direction, not just in the direction of the axes.

The Attempt at a Solution

Since the triangle is isoceles. |u|=|v|
u.v=(sqrt(3)+sqrt(3) )cos60=-3.29925

Is this correct?

 

[knowLittle]

I thought that the vectors shown in my graph could be considered as forming a triangle.

The graph is attached, after the first post.

 

[Mark44]

I'm confused. Your drawing shows tickmarks on all three sides, which implies that all three are equal in length and that the triangel is equilateral. In post #1, you say that the triangle is isosceles, which implies that only two sides are equal. Which is it?

 

[knowLittle]

I'm sorry, it's an equilateral triangle. The graph is correct. The problem does not state anything other than what is shown in the graph and the first post.

 

[Mark44]

Your sketch is NOT correct if the triangle is only isosceles. The tickmarks on all thee sides imply that it is equilateral, which is different from being isosceles.

Are you given that all three sides are equal in length? If so, then stating that the triangle is isosceles confuses the matter.

 

[knowLittle]

You are right, it's equilateral.

 

[Mark44]

OK, we're finally getting somewhere. If u is a unit vector, then so are v and w.

Another formulation for the dot product is u$\cdot$v = |u||v|cosθ. In this case, θ is 60°, as are the other two interior angles.

So what do you get for u$\cdot$v and u$\cdot$w?

 

[knowLittle]

u.v= (u^2)cos60=u^2(1/2)

u.w=(u^2)cos60=(u^2)1/2

 

[Mark44]

u.v= (u^2)cos60=u^2(1/2)

u.w=(u^2)cos60=(u^2)1/2

No. Do you understand what the term unit vector means?

Also, what does u² even mean for a vector?

 

[knowLittle]

I wrote u^2, but I meant |u^2|. I realize that there's no multiplication of vectors.

I don't think I know, what a unit vector means. I know that a unit vector is (vector/|vector|), but that's it.

 

[Mark44]

I wrote u^2, but I meant |u^2|. I realize that there's no multiplication of vectors.

That still doesn't make any sense, because you are multiplying a vector by itself, and then taking the magnitude of the result.

This makes more sense: |u|², the square of the magnitude of vector u.

I don't think I know, what a unit vector means. I know that a unit vector is (vector/|vector|), but that's it.

If there's something you don't know, your very first impulse should be to find out how it is defined. Math books are careful to define the terms they use very precisely. A unit vector is any vector whose length is 1. That's why they call it a unit vector - the word "unit" implies "oneness".