Dot product of $$$\left\langle 4, -2, -4\right\rangle$$$ and $$$\left\langle 1, 2, -2\right\rangle$$$

The calculator will find the dot product of two vectors $$$\left\langle 4, -2, -4\right\rangle$$$ and $$$\left\langle 1, 2, -2\right\rangle$$$, with steps shown.
$$$\langle$$$ $$$\rangle$$$
$$$\langle$$$ $$$\rangle$$$

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Your Input

Calculate $$$\left\langle 4, -2, -4\right\rangle\cdot \left\langle 1, 2, -2\right\rangle$$$.


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: $$$\left\langle 4, -2, -4\right\rangle\cdot \left\langle 1, 2, -2\right\rangle = \left(4\right)\cdot \left(1\right) + \left(-2\right)\cdot \left(2\right) + \left(-4\right)\cdot \left(-2\right) = 8.$$$


$$$\left\langle 4, -2, -4\right\rangle\cdot \left\langle 1, 2, -2\right\rangle = 8$$$A