Question

For the given vectors A = a x + a y +2a z and B = 2a x - a y + a z , A.B is

Answer 1

The dot product of two vectors in $\mathbb{R}^3$ is defined as $\bold{a} . \bold{b} = a_x b_x + a_y b_y + a_z b_z$.

Hence, for vectors $\bold a = (a, a, 2 a)$ and $\bold b = (2 a, -a, a)$:

$\bold a . \bold b = a \cdot 2 a - a \cdot a + 2a \cdot a = 3 a^2$.