Dot product of $$$\left\langle 2, -3, 1\right\rangle$$$ and $$$\left\langle 4, -1, -3\right\rangle$$$
Your Input
Calculate $$$\left\langle 2, -3, 1\right\rangle\cdot \left\langle 4, -1, -3\right\rangle$$$.
Solution
The dot product is given by $$$\mathbf{\vec{u}}\cdot \mathbf{\vec{v}} = \sum_{i=1}^{n} u_{i} v_{i}$$$.
Thus, what we need to do is multiply the corresponding coordinates and then add up the results: $$$\left\langle 2, -3, 1\right\rangle\cdot \left\langle 4, -1, -3\right\rangle = \left(2\right)\cdot \left(4\right) + \left(-3\right)\cdot \left(-1\right) + \left(1\right)\cdot \left(-3\right) = 8.$$$
Answer
$$$\left\langle 2, -3, 1\right\rangle\cdot \left\langle 4, -1, -3\right\rangle = 8$$$A