Dot product of $$$\left\langle 1, 1, 3\right\rangle$$$ and $$$\left\langle 0, 3, 4\right\rangle$$$

The calculator will find the dot product of two vectors $$$\left\langle 1, 1, 3\right\rangle$$$ and $$$\left\langle 0, 3, 4\right\rangle$$$, with steps shown.
$$$\langle$$$ $$$\rangle$$$
Comma-separated.
$$$\langle$$$ $$$\rangle$$$
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Your Input

Calculate $$$\left\langle 1, 1, 3\right\rangle\cdot \left\langle 0, 3, 4\right\rangle$$$.

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: $$$\left\langle 1, 1, 3\right\rangle\cdot \left\langle 0, 3, 4\right\rangle = \left(1\right)\cdot \left(0\right) + \left(1\right)\cdot \left(3\right) + \left(3\right)\cdot \left(4\right) = 15.$$$

Answer

$$$\left\langle 1, 1, 3\right\rangle\cdot \left\langle 0, 3, 4\right\rangle = 15$$$A