Dot product

Notes:

Dot Product
Let $u=<a,b,c>$ and $v=<d,e,f>$. Then the dot product of these two vectors will be:

$u \cdot v = ad + be + df$

Dot Product Property
If the dot product of two vectors $u$ and $v$ gives 0, then the vectors are perpendicular. In other words,

$u \cdot v=0 \to$ perpendicular vectors

Scalar and Vector Projection
Suppose we have two vectors $a$ and $b$. Suppose they create an angle $\theta$ such that we get the following picture:

$|v| = \frac{a \cdot b } {|b|}$

To find the vector projection $a$ onto $b$ (which is v), we use the formula:

$v = \frac{a \cdot b}{b \cdot b}b$

Additional Dot Product Properties
Let $u, v, w$ be vectors and $c$ be a scalar. Then the properties of dot products are:

1. $u \cdot u = |u|^2$
2. $u \cdot v = v \cdot u$
3. $u \cdot (v+w) = u \cdot v + u \cdot w$
4. $(cu) \cdot v = u \cdot (cv) = c(u \cdot v)$