Dot product of $$$\left\langle 3, 4, 5\right\rangle$$$ and $$$\left\langle 4, -5, -3\right\rangle$$$
Your Input
Calculate $$$\left\langle 3, 4, 5\right\rangle\cdot \left\langle 4, -5, -3\right\rangle$$$.
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: $$$\left\langle 3, 4, 5\right\rangle\cdot \left\langle 4, -5, -3\right\rangle = \left(3\right)\cdot \left(4\right) + \left(4\right)\cdot \left(-5\right) + \left(5\right)\cdot \left(-3\right) = -23.$$$
Answer
$$$\left\langle 3, 4, 5\right\rangle\cdot \left\langle 4, -5, -3\right\rangle = -23$$$A