Main Concept

Given two vectors  $\stackrel{\mathbf{\rightharpoonup }}{\mathit{a}}$ and $\stackrel{\mathbf{\rightharpoonup }}{\mathit{b}}$, their dot product is the scalar quantity

The dot product can also be expressed in terms of the components of $\stackrel{\rightharpoonup }{a}$ and $\stackrel{\rightharpoonup }{b}$ as follows:

$\stackrel{\rightharpoonup }{a}\cdot \stackrel{\rightharpoonup }{b }\= a_1\cdot b_1 \+ a_{2 }\cdot b_2 \+a_3\cdot b_3$$$

The unit vector in the direction of  $\stackrel{\rightharpoonup }{a}$ is given by

$\stackrel{\^}{a}\=\frac{\stackrel{\rightharpoonup }{a}}{\left|a\right|}$

The vector projection of  $\stackrel{\rightharpoonup }{a}$ on $\stackrel{\rightharpoonup }{b}$ is the orthogonal projection of $\stackrel{\rightharpoonup }{a}$ onto the line in the direction of $\stackrel{\rightharpoonup }{b}$:

$\left( \left|a\right|\cdot \cos \left(\mathrm{\theta }\right)\right)\cdot \stackrel{\^}{b} \=\left( \stackrel{\rightharpoonup }{a}\cdot \stackrel{\^}{b}\right)\cdot \stackrel{\^}{b} \=\frac{\left(\stackrel{\rightharpoonup }{a}\cdot \stackrel{\rightharpoonup }{b }\right)\cdot \stackrel{\rightharpoonup }{b }}{{\left|b\right|}^2}\= \frac{a_x\cdot b_x \+ a_y \cdot b_y \+ a_z\cdot b_z}{b_x^2\+ b_y^2 \+{\mathrm{b}}_y^2} \cdot \stackrel{\rightharpoonup }{b }$

The scalar projection of  $\stackrel{\rightharpoonup }{a}$ on $\stackrel{\rightharpoonup }{b}$ is the length of the associated vector projection.

$\left|a\right|\cdot \cos \left(\mathrm{\theta }\right) \= \stackrel{\rightharpoonup }{a}\cdot \stackrel{\^}{b} \=\frac{\stackrel{\rightharpoonup }{a}\cdot \stackrel{\rightharpoonup }{b }}{\|b\|} \= \frac{a_x\cdot b_x \+ a_y \cdot b_y \+ a_z\cdot b_z}{\sqrt{b_x^2\+ b_y^2 \+{\mathrm{b}}_y^2}}$

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