# Dot Product (Geometric)

Vector multiplication can be completed using the dot product or cross product. The result of the dot product is a scalar. We can calculate the dot product using geometric or algebraic vectors. The geometric dot product of two vectors $$\vec{a}$$ and $$\vec{b}$$ where θ is the angle between the two vectors is defined by:

$$\vec{a}\cdot\vec{b} = \vert \vec{a} \vert \vert \vec{b} \vert \cos{θ}$$

### Dot Product Properties

$$\vec{u}\cdot\vec{u}=0$$ iff $$\vec{u}\perp\vec{v}$$

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

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

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

$$\vec{u}\cdot\left(\vec{a}+\vec{b}\right)=\vec{u}\cdot\vec{a}+\vec{u}\cdot\vec{b}$$

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

Find the dot product for $$\vec{u}\cdot\vec{v}$$ for each of the following where θ is the angle between vectors.
$$\vert\vec{u}\vert=7$$, $$\vert\vec{v}\vert=12$$, $$θ=60°$$
$$\vert\vec{a}\vert=20$$, $$\vert\vec{b}\vert=3$$, $$θ=\dfrac{5\pi}{6}$$

For the above question find $$\vec{v}\cdot\vec{u}$$. What is property you can conclude from this?

Find $$\vec{a}\cdot\vec{a}$$ and $$\vec{b}\cdot\vec{b}$$. What can you conclude from this?

Find $$\vec{a}\cdot\vec{0}$$ and $$\vec{v}\cdot\vec{0}$$. What can conclude from this?

Prove that two non-zero vectors $$\vec{u}$$ and $$\vec{v}$$ are perpendicular, if and only if $$\vec{u}\cdot\vec{v}=0$$.

Proof 1:

Proof 2:

Explain why $$\left(\vec{a}\cdot\vec{b}\right)\cdot\vec{c}\neq\vec{a}\cdot\left(\vec{c}\cdot\vec{b}\right)$$