Dot Product

Topics:

• Know how to compute the dot product of two vectors.
• Be able to use the dot product to find the angle between two vectors; and, in particular, be able to determine if two vectors are orthogonal.
• Know how to compute the direction cosines of a vector.
• Be able to decompose vectors into orthogonal components.
  • Know how to compute the orthogonal projection of one vector onto another.

The dot (or inner or scalar) product of $v=<v_1,v_2>$ and $w=<w_1,w_2>$ is defined to be

$$\vec{v}\cdot \vec{w}:=v_1{w}_1+v_2{w}_2$$

Or, if in 3D

$$\vec{v}\cdot \vec{w}:=v_1{w}_1+v_2{w}_2+v_3{w}_3$$

Dot Product Properties Theorem. If $u$, $v$ and $w$ are vectors in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ and $k$ is a scalar, then:

$\vec{u}\cdot \vec{v}=\vec{v}\cdot \vec{u}$

$\vec{u}\cdot (\vec{v}+\vec{w})=(\vec{u}+\vec{v})+(\vec{u}+\vec{w})$

$k(\vec{u}\cdot \vec{v})=\vec{v}\cdot (k\vec{u})=\vec{u}\cdot (k\vec{v})$

$0\cdot \vec{u}=0$

and most importantly

$\vec{u}\cdot \vec{u}=\Vert \vec{u}{\Vert }^2$

Geometrically…

$\vec{v}\cdot \vec{w}:=\Vert \vec{v}\Vert \Vert \vec{w}\Vert \cos \Theta$

BY THE LAW OF COSINES

$\Vert \vec{u}-\vec{v}{\Vert }^2=\Vert \vec{u}{\Vert }^2+\Vert \vec{v}{\Vert }^2-2\Vert \vec{u}\Vert \Vert \vec{v}\Vert cos\Theta$

$(\vec{u}-\vec{v})\cdot (\vec{u}-\vec{v})=\vec{u}\cdot \vec{u}+\vec{v}\cdot \vec{v}2\Vert \vec{u}\Vert \Vert \vec{v}\Vert cos\Theta$

Thus

$\Theta =\arccos (\frac{\vec{u}\cdot \vec{v}}{\Vert \vec{u}\Vert \Vert \vec{v}\Vert })$

omg! we can find the angle between two vectors without drawing them… waoooow

Direction Cosines

We can refer to a vector in 2-space with its magnitude and cosine??

In 3-space we can specify the direction of a vector by giving the angles $\alpha$, $\beta$ and $\gamma$ between v and the unit vectors $\vec{i}$, $\vec{j}$ and $\vec{k}$ respectively: these are the direction cosines of v

When we normalize a vector $v$, the components of the resulting unit vector are the three direction cosines

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