Thread: Proving Vector Dot Product Properties

I have a big hole in my math skills in that I don't really no where to begin in proving properties. I can use the dot product to work out things just not sure how to prove the properties.

Let U = (Ux, Uy, Uz), Let V = (Vx, Vy, Vz), Let W = (Wx, Wy, Wz), Let c and k be scalars. Prove the following dot properties.

A. U . V = V . U
B. U . (V + W) = U . V + U . W
C. k (U . V) = (kU) . V = U . (kV)
D. V . V = || V ||^2
E. 0 . V = 0

I have worked out D as:-

V . V = Vx Vx + Vy Vy + Vz Vz
= Vx^2 + Vy^2 + Vz^2
= ( SQRT( Vx^2 + Vy^2 + Vz^2) )^2
= ( ||V|| )^2

With A I know that,

U . V = Ux Vx + Ux Vy + Uz Vz
V . U = Vx Ux + Vy Uy + Vz Uz

I just don't know how to prove U . V = V . U without using values. My problem is the same for the remaining questions.

What topic do I need to study further to improve this skill I'm lacking?

Originally Posted by RedKMan

I just don't know how to prove U . V = V . U without using values.

Just notice that .
That is true for the other two components as well.
Multiplication is commutive.

So I could simply put:-

(Ux Vx + Uy Vy + Uz Vz) = (Vx Ux + Vy Uy + Vz Uz) = U . V = V . U

This would be enough to prove that U . V = V . U ?

Originally Posted by RedKMan

So I could simply put:-

(Ux Vx + Uy Vy + Uz Vz) = (Vx Ux + Vy Uy + Vz Uz) = U . V = V . U

This would be enough to prove that U . V = V . U ?

Yes it is exactly that.
U . V =(Ux Vx + Uy Vy + Uz Vz) = (Vx Ux + Vy Uy + Vz Uz) = V . U

For this property, U . (V + W) = U . V + U . W

I have,

U . ( V + W ) = ( Ux(Vx + Wx) + Uy (Vy + Wy) + Uz (Vz + Wz)

= (Ux Vx + Uy Vy + Uz + Vz) + (Ux Wx + Uy Wy + Uz Wz) = U . V + U.W

Do I have the right idea here?

Originally Posted by RedKMan

For this property, U . (V + W) = U . V + U . W

I have,

U . ( V + W ) = ( Ux(Vx + Wx) + Uy (Vy + Wy) + Uz (Vz + Wz)

= (Ux Vx + Uy Vy + Uz + Vz) + (Ux Wx + Uy Wy + Uz Wz) = U . V + U.W

Do I have the right idea here?

Yes, that works nicely.
If would be nice if you could learn to use LaTex.

[tex](U_x V_x + U_y V_y + U_z V_z) + (U_x W_x + U_y W_y + U_z W_z) = U\cdot V +U\cdot W[/tex]
gives

I just wanted to confirm this is right:-

Originally Posted by RedKMan

I just wanted to confirm this is right:-

This is correct.

You have too many k's in

That has confused me a bit as I thought the scalar would apply to both the and the vectors. I thought both vectors would be multiplied by the scalar k.

I was working from the idea that 2(a + b) = (2 * a) + (2 * b).

Originally Posted by RedKMan

That has confused me a bit as I thought the scalar would apply to both the and the vectors. I thought both vectors would be multiplied by the scalar k.
I was working from the idea that 2(a + b) = (2 * a) + (2 * b).