Evaluate and simplify the quantity. $$\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle$$
A) $t ^ { 2 } + e ^ { 2 t }$
B) $2 t e ^ { 2 t }$
C) 0
D) $- 2 t e ^ { t }$

Question

Quizplus · Accepted Answer

The dot product of two vectors $\left\langle a, b \right\rangle$ and $\left\langle c, d \right\rangle$ is calculated as $ac + bd$. Applying this to the given vectors, we get $(-t \cdot e^t) + (e^t \cdot -t) = -te^t - te^t = -2te^t + 2te^t = 0$.

Evaluate and Simplify the Quantity ⟨t,et⟩⋅⟨−et,−t⟩\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle⟨t,et⟩⋅⟨−et,−t⟩

Evaluate and Simplify the Quantity $\left\langle t , e ^ { t } \right\rangle \cdot \left\langle - e ^ { t } , - t \right\rangle$