Using the product i would like to understand if it is possible to determine relationship between two vectors,let us consider the following problem:
Vectors $u$ and $v$ have length $1$. Which of the assumptions $(a)-(g)$ below imply that vectors $u$ and $v$ are: $$\begin{array}{c|c}
\text{i} & \text{prependicular} \\
\hline
\text{ii} & \text{parallel and pointing in the same direction} \\
\hline
\text{iii} & \text{parallel and pointing in the opposite direction} \\
\hline
\text{iv} & \text{parallel and no information about directions}
\end{array}$$ $$\text{(a) } u \cdot v = -1$$ $$\text{(b) } u \cdot v = 1$$ $$\text{(c) } |u \cdot v| = 1$$ $$\text{(d) } |u \cdot v| = 0$$ $$\text{(e) } u \cdot v = 0$$
Sure one thing we can easily say that if dot product of two vector is zero,they are perpendicular (orthogonal),parallel and pointing in the same direction,for example if we multiply vector by some scalar ,we get parallel and same direction vector,maybe product of them should be positive?if opposite direction,maybe their product is $-1$ ?about $(iv)$ maybe if absolute value of product is $1$
Best Answer
$\mathbf u. \mathbf v=|\mathbf u||\mathbf v|\cos\theta$ where $\theta$ is the angle between vectors. So If they are parallel and pointing to the same direction then $\mathbf u. \mathbf v=1$, parallel and pointing to the opposite direction then $\mathbf u. \mathbf v=-1$, parallel and no information about the direction $|\mathbf u. \mathbf v|=1$.