Dot product of $$$\left\langle 1, 2 t, 3 t^{2}\right\rangle$$$ and $$$\left\langle 0, 2, 6 t\right\rangle$$$

Calculate $$$\left\langle 1, 2 t, 3 t^{2}\right\rangle\cdot \left\langle 0, 2, 6 t\right\rangle$$$.

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: $$$\left\langle 1, 2 t, 3 t^{2}\right\rangle\cdot \left\langle 0, 2, 6 t\right\rangle = \left(1\right)\cdot \left(0\right) + \left(2 t\right)\cdot \left(2\right) + \left(3 t^{2}\right)\cdot \left(6 t\right) = 18 t^{3} + 4 t.$$$

$$$\left\langle 1, 2 t, 3 t^{2}\right\rangle\cdot \left\langle 0, 2, 6 t\right\rangle = 18 t^{3} + 4 t$$$A