PlanetPhysics/Dot Product Example

Examples involving the dot product:

(1) Calculate $$ \mathbf{A} \cdot \mathbf{B} $$ with

$$ \mathbf{A} = 5 \mathbf{\hat{i}} + 2 \mathbf{\hat{j}} - \mathbf{\hat{k}}$$ \\ $$ \mathbf{B} = 3 \mathbf{\hat{i}} - 3 \mathbf{\hat{j}} + 3\mathbf{\hat{k}}$$ \\

answer:

$$ \mathbf{A} \cdot \mathbf{B} = (5)(3) + (2)(-3) + (-1)(+3) $$ \\ $$ \mathbf{A} \cdot \mathbf{B} = 15 - 6 - 3 = 6$$ \\

(2) Find the angle between the above vectors.

answer:

We know their dot product, so we just need to calculate their magnitudes $$ \left | \mathbf{A} \right | = \sqrt{A_x^2 + A_y^2 + A_z^2} = \sqrt{5^2 + 2^2 +(-1)^2} $$ \\ $$ \left | \mathbf{A} \right | = \sqrt{25 + 4 + 1} = \sqrt{30} = 5.48 $$ \\ \\

$$ \left | \mathbf{B} \right | = \sqrt{3^2 + (-3)^2 +(3)^2} $$ \\ $$ \left | \mathbf{B} \right | = \sqrt{9 + 9 + 9} = \sqrt{27} = 5.2 $$ \\ \\

Finally

$$ \mathbf{A} \cdot \mathbf{B} = \left | \mathbf{A} \right | \left | \mathbf{B} \right | \cos \theta$$

$$ \theta = cos^{-1} \left ( \frac{ \mathbf{A} \cdot \mathbf{B} }{ \left | \mathbf{A} \right | \left | \mathbf{B} \right |} \right )$$ \\

$$ \theta = cos^{-1} \left ( \frac{ 6 }{ (5.48)(5.2)} \right) $$ \\ $$ \theta = cos^{-1} (0.21) = 77.9^o $$