For A = 5î - 2ĵ and B = 3î + 4ĵ, find A · B and the angle between the vectors.

Step 1. Dot product. $\vec{A} \cdot \vec{B} = 5 \times 3 + (-2) \times 4 = 15 - 8 = 7$.

Step 2. Sizes. $A = \sqrt{25 + 4} = \sqrt{29} = 5.39$ and $B = \sqrt{9 + 16} = 5$.

Step 3. Angle. $\cos\theta = \dfrac{7}{5.39 \times 5} = 0.260$, so $\theta = 74.9^\circ$.

Step 4. (b) $\vec{P} \cdot \vec{Q} = 12 - 12 = 0$, so yes, they are at right angles.