Calculate the dot product of vectors A = 3i + 4j − k and B = -2i + 3j + k.

Given vectors are:

$$A = 3i + 4j - k$$ $$B = -2i + 3j + k$$

The components are:
For A: $A_x = 3$, $A_y = 4$, $A_z = -1$
For B: $B_x = -2$, $B_y = 3$, $B_z = 1$

The dot product of two vectors $A = A_x i + A_y j + A_z k$ and $B = B_x i + B_y j + B_z k$ is given by:

$$A \cdot B = A_x B_x + A_y B_y + A_z B_z$$

Substitute the components:

$$A \cdot B = (3)(-2) + (4)(3) + (-1)(1)$$ $$A \cdot B = -6 + 12 - 1$$ $$A \cdot B = 6 - 1$$ $$A \cdot B = 5$$

The correct option is A) 5.