Dot product of $$$\left\langle 5, -2, 3\right\rangle$$$ and $$$\left\langle -4, 5, 7\right\rangle$$$
Your Input
Calculate $$$\left\langle 5, -2, 3\right\rangle\cdot \left\langle -4, 5, 7\right\rangle$$$.
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: $$$\left\langle 5, -2, 3\right\rangle\cdot \left\langle -4, 5, 7\right\rangle = \left(5\right)\cdot \left(-4\right) + \left(-2\right)\cdot \left(5\right) + \left(3\right)\cdot \left(7\right) = -9.$$$
Answer
$$$\left\langle 5, -2, 3\right\rangle\cdot \left\langle -4, 5, 7\right\rangle = -9$$$A