Vector Review

A vector $\vec{v}$ is a directed line segment that has length and direction.

• Vector equality: Two vectors are equal when they have same length and direction

Definition: Standard basis in ${\mathbb{R}}^3$

Standard basis vectors for ${\mathbb{R}}^3$ are

$$\vec{i}=⟨1,0,0⟩,\vec{j}=⟨0,1,0⟩,\vec{k}=⟨0,0,1⟩$$

All vectors in ${\mathbb{R}}^3$ can be written in terms of $\vec{i},\vec{j},\vec{k}$.

Unit vector: Vectors with length 1 Normalization of vectors: To find a unit vector going in the direction of a given vector, take

$$\vec{v}=\frac{\vec{v}}{\mid \mid \vec{v}\mid \mid }$$

Inner Product (dot product)

The standard dot product of two vectors $\vec{u}=⟨u_1,u_2,u_3⟩$ and $\vec{v}=⟨v_1,v_2,v_3⟩$ is:

$$\vec{u}\cdot \vec{v}=u_1{v}_1+u_2{v}_2+u_3{v}_3$$

The angle between two vectors can be found with:

$$\vec{u}\cdot \vec{v}=\mid \mid \vec{u}\mid \mid \mid \mid \vec{v}\mid \mid \cos \theta$$

The two vectors $\vec{u}$ and $\vec{v}$ are perpendicular iff $\vec{u}\cdot \vec{v}=0$.

Cross Product

$$\vec{u}\times \vec{v}=\left|\begin{array}{ccc}\vec{i}& \vec{j}& \vec{k}\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right|=(u_2{v}_3-u_3{v}_2)\vec{i}-(u_1{v}_3-u_3{v}_1)\vec{j}+(u_1{v}_2-u_2{v}_1)\vec{k}$$

The cross-product is anti-communicative, such that:

$$\vec{u}\times \vec{v}=-\vec{v}\times \vec{u}$$

It is also distributive, such that:

$$\vec{u}\times (a\vec{v}+b\vec{w})=a(\vec{u}\times \vec{v})+b(\vec{u}\times \vec{w})$$

The angle between two vectors can be found with:

$$\mid \mid \vec{u}\times \vec{v}\mid \mid =\mid \mid \vec{u}\mid \mid \mid \mid \vec{v}\mid \mid \sin \theta$$