Overview

Vectors are have two properties, #Magnitude and direction

A vector $\vec{P Q}$ can be found from two points $P$ and $Q$ by subtracting $P$ from $Q$ .

Special Vectors

The zero-vector has no magnitude and no direction. in $R^{2}$ and $R^{3}$

A unit vector is a vector with #magnitude of 1 or

| | \vec{v} | | = 1

Given a vector $\vec{v}$ the unit vector can be found using the following:

\frac{\vec{v}}{∥ \vec{v} ∥}

Standard Basis Vectors

In $R^{2}$ : $i = ⟨ 1, 0 ⟩$ and $j = ⟨ 0, 1 ⟩$

In $R^{3}$ : $i = ⟨ 1, 0, 0 ⟩$ and $j = ⟨ 0, 1, 0 ⟩$ and $j = ⟨ 0, 0, 1 ⟩$

Magnitude

The magnitude or length of a vector can be found in one of the following ways.

Notation: $∥ v ∥$ meaning the magnitude of vector v

e.g. If the vector is in $R^{2}$ and $\vec{v} = ⟨ 2, 3 ⟩$ :

\begin{aligned} ∥ \vec{v} ∥ & = \sqrt{2^{2} + 3^{2}} \\ = \sqrt{4 + 9} \\ = \sqrt{13} \end{aligned}

Or similarly if the vector is in $R^{3}$ and $\vec{w} = ⟨ 2, 3, 4 ⟩$ :

\begin{aligned} ∥ \vec{v} ∥ & = \sqrt{2^{2} + 3^{2} + 4^{2}} \\ = \sqrt{4 + 9 + 16} \\ = \sqrt{29} \end{aligned}

Dot Product

A dot product is an operation that is performed on two different vectors which produces a constant or scalar.

If the dot product of two vectors is $0$ the angle between them is orthogonal.

The dot product has the following properties:

Communitive | $u \cdot v = v \cdot u$
Zero | $0 \cdot v = 0$
Distributive | $u \cdot (v + w) = u \cdot v + u \cdot w$
Square | $u \cdot u = ∥ u ∥^{2}$
Constant Multiple | $c (u \cdot v) = c v \cdot u = v \cdot c u$

Calculations

The dot product of two vectors is found as following in $R^{2}$ :

⟨ i_{1}, j_{1} ⟩ \cdot ⟨ i_{2}, j_{2} ⟩ = i_{1} * i_{2} + j_{1} * j_{2}

Or in $R^{3}$ :

⟨ i_{1}, j_{1}, k_{1} ⟩ \cdot ⟨ i_{2}, j_{2}, k_{2} ⟩ = i_{1} * i_{2} + j_{1} * j_{2} + k_{1} * k_{2}

e.g. Given $v = ⟨ 1, 2, 3 ⟩$ and $w = ⟨ 4, 5, 6 ⟩$

\begin{aligned} v \cdot w & = 1 * 4 + 2 * 5 + 3 * 6 \\ = 4 + 10 + 18 \\ = 32 \end{aligned}

The following formula can be manipulated to find the angle between two vectors:

\cos (θ) = \frac{u \cdot v}{∥ u ∥ ∥ v ∥}

Cross Product

A cross product is an operation performed between two different vectors which produces a resulting vector.

The cross product of two vectors is always orthogonal to the two original vectors.

Calculations

The cross product of two vectors can be found as following in $R^{3}$ :

⟨ i_{1}, j_{1}, k_{1} ⟩ \times ⟨ i_{2}, j_{2}, k_{2} ⟩ = ⟨ j_{1} * i_{2} - k_{1} * j_{2}, k_{1} * i_{2} - i_{1} * k_{2}, i_{1} * j_{2} - j_{1} * i_{2} ⟩

The following formula can be manipulated to find the angle between two vectors

\sin θ = \frac{∥ u \times v ∥}{∥ u ∥ ∥ v ∥}

Lines

Lines $R^{3}$ are made up of a direction vector $\vec{v}$ and a point $P$ .

Given $\vec{v} = ⟨ A, B, C ⟩$ and point $P = (a, b, c)$

r (t) = ⟨ A t + a, B t + b, C t + c ⟩

r (t) = ⟨ a, b, c ⟩ + t ⟨ A, B, C ⟩

Lines are parallel if the direction vector of one line can be multiplied by a scalar to get the direction vector of the other.

e.g. Given two lines $r (t) = ⟨ 3 t + 1, 7 t + 13, 11 t + 8 ⟩$ and $s (t) = ⟨ 30 t + 1, 70 t + 13, 110 t + 8 ⟩$

Lines $r$ and $s$ are parallel because each component of the direction vector of $r = ⟨ 3, 7, 11 ⟩$ can be multiplied by $10$ to get the direction vector of $s = ⟨ 30, 70, 110$ .

Definitions

Linear Combinations
#linear-combo

A linear combination of points/Vectors/columns

Is (informally) any point/vector/column that can be constructed with using point ( $x_{1}$ , $x_{n}$ ) and ( $y_{1}$ , $y_{n}$ )

(formally) any point/vector/column of the form $α x + β y$ where $α$ and $β$ are constants.

e.g.

Given the points $a = (1, 1)$ and $b = (2, 3)$ any point on the line created by points $a$ and $b$ would be a linear combo. Any point outside the line would not be a linear combo.