Check out some similar questions!

brijmohan123456 · Oct 17, 2012, 06:56 AM

Can anyone give me proof and complete explanation to the fact that
"the dot product of vector A with a unit vector B is the magnitude of A in the direction B."
Please do not get annoyed by my question.
Thank you

ebaines · Oct 18, 2012, 07:18 AM

The dot product of vectors $\vec A$ and $\vec B$ is defined as the magnitude of A times the magnitude of B times the cosine of the angle between them. If $\vec B$ is a unit vector, then its magnitude is 1, and the dot product of $\vec A$ with the unit vector is equal to the magnitude of A times the cosine of the angle to the unit vector.

The vector $\vec A$ can be decomposed into two orthognal vectors - call them $\vec {A_1}$ and $\vec {A_2}$ with angle $\theta$ between $\vec A$ and $\vec {A_1}$ . The length of $\vec {A_1}$ is therefore $|A| \cos \theta$ , which is equal to the length of the projection of $\vec A$ in the $\vec {A_1}$ direction. The length of $\vec A_1$ is therefore same as the dot product of $\vec A$ and the unit vector in the $\vec A_1$ direction.

Hope this helps!