dot product examples | Physics Library

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 $\left \vert \mathbf{A} \right \vert = \sqrt{A_x^2 + A_y^2 + A_z^2} = \sqrt{5^2 + 2^2 +(-1)^2}$
$\left \vert \mathbf{A} \right \vert = \sqrt{25 + 4 + 1} = \sqrt{30} = 5.48$

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

Finally

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

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

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

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