# Inner product, length, and orthogonality

##### Intros
###### Lessons
1. Inner Product, Length, and Orthogonality Overview:
2. The Length of a Vector
• Also known as norm
• Some properties with a length of a vector
• Unit vectors
3. Distance of two vectors/Orthogonal Vectors
• Dist$(u,v)=\lVert u-v \rVert$
• How to tell with two vectors are orthogonal
• Orthogonal vectors with Pythagorean Theorem
##### Examples
###### Lessons
1. Utilizing the Inner Product
Given that and , compute:
$\frac{u \cdot u}{v \cdot v}v$
1. Given that and , compute:
$\lVert (u \cdot v)u \rVert$
1. Finding the Unit vector
Find the unit vector in the direction of the vector
1. Calculating the Distance
Find the distance between the vectors and .
1. Showing Orthogonality
For what value(s) of $b$ make vectors $u$ and $v$ orthogonal if
1. Proof Questions
Show that the parallelogram is true for vectors $u$ and $v$ in R^n. In other words, show that:
$\lVert u+v \rVert^2 + \lVert u-v \rVert^2 = 2 \lVert u \rVert^2 + 2 \lVert v \rVert^2$
1. Suppose vector $u$ is orthogonal to vectors $x$ and $y$. Show that $u$ is also orthogonal to $x+y$.