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[parent] dot product examples (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 \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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Cross-references: magnitudes, vectors, dot product

This is version 2 of dot product examples, born on 2006-07-22, modified 2006-07-22.
Object id is 207, canonical name is DotProductExample.
Accessed 2857 times total.

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Physics Classification: 02. (Mathematical methods in physics)
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