vector dot product question

**duaneg37** · February 2nd 2011, 12:23 PM

If I define vectors $\vec{CA}=\vec{V}+\vec{U}$ and $\vec{CB}=\vec{V}-\vec{U}$ is the following true?

$\vec{CA}\cdot\vec{CB}=\vec{V}^{2}-\vec{U}^{2}$

thanks for the help!

**Ackbeet** · February 2nd 2011, 12:26 PM

The dot product obeys the usual foiling rule for multiplying out. In addition, the dot product is commutative. Those two facts should enable you to tell me the answer to your question.

**duaneg37** · February 2nd 2011, 12:34 PM

I thought the dot product gave a number, not a vector. So I am not sure.

**Ackbeet** · February 2nd 2011, 12:36 PM

Originally Posted by duaneg37

I thought the dot product gave a number, not a vector. So I am not sure.

It does. But the square of a vector (the square of its length) is also a number. So there at least a fighting chance that you're right. Show me what you get when you foil out the dot product in the usual way.

**Plato** · February 2nd 2011, 12:42 PM

You are correct. The dot product is a scalar.
I think your mistake is in notation.
$\vec{CA}\cdot\vec{CB}=\vec{V}\cdot\vec{V}-\vec{U}\cdot\vec{U}$ . Do see why that is the case?
Now $\vec{V}\cdot\vec{V}=\|\vec{V}\|^2$ , the length squared.
So we have $\|\vec{V}\|^2-\|\vec{U}\|^2$ , a number.

**Ackbeet** · February 2nd 2011, 12:44 PM

Reply to Plato:

Perhaps. But I've seen notation abused in this manner all the time, especially by physicists.

**Plato** · February 2nd 2011, 12:47 PM

Originally Posted by Ackbeet

Reply to Plato:
Perhaps. But I've seen notation abused in this manner all the time, especially by physicists.

I did not see your reply before posting. Sorry.
But I think it is still an important point.

**duaneg37** · February 2nd 2011, 01:01 PM

Suppose that $AB$ is the diameter of a circle with center $O$ , and that $C$ is a point on one of the two arcs joining $A$ and $B$ . Show that $\vec{CA}$ and $\vec{CB}$ are orthogonal. $\vec{V}$ is from $O$ to $C$ . $\vec{U}$ is from $O$ to $B$ . $\vec{-U}$ is from $O$ to $A$ .

I know if their dot product is zero, they are orthogonal. If the dot product is $\vec{U}^{2}-\vec{V}^{2}$ it would have to be zero because their magnitudes are the same.

Is this enough proof? I think it is.

**Ackbeet** · February 2nd 2011, 01:06 PM

Had to think about it for a while, but yes, I think your proof works fine.

Thread: vector dot product question

LinkBack

Thread Tools

Display

vector dot product question

Similar Math Help Forum Discussions

Finding a vector given another vector and their cross product

How do I do vector product / scalar product using maxima?

Gr 12. Vector Question Involving Dot Product of Two Vectors(Part 3)

Gr 12. Vector Question Involving Dot Product of Two Vectors

Gr 12. Vector Question Involving Dot Product of Two Vectors(Part 2)

Search Tags