# Vectors

A vector is a quantity that has both a magnitude and a direction.

## Vector and scalar quantities

A scalar quantity is completely specified by a single value with an appropriate unit and has no direction.

A vector quantity is completely specified by a number with an appropriate unit (the magnitude of the vector) plus a direction.

## Notation

For representing a vector, the recommended notation is non-bold italic serif accented by a right arrow: $\vec{v}$.

In advanced mathematics, vectors are often represented in a simple italic type.

### Rectangular vectors

#### Ordered set notation

A rectangular vector in $\mathbb{R}^n$ can be specified using an ordered set of components, enclosed in either parentheses or angle brackets.

• $\vec{v} = (v_{1}, v_{2}, ... v_{n})$
• $\vec{v} = \langle v_{1}, v_{2}, ... v_{n} \rangle$

#### Matrix notation

A rectangular vector in $\mathbb{R}^n$ can be specified as a row or column matrix containing the ordered set of components.

• $\vec{v} = \begin{bmatrix} v_{1} & v_{2} & ... & v_{n} \end{bmatrix} = \begin{pmatrix} v_{1} & v_{2} & ... & v_{n} \end{pmatrix}$
• $\vec{v} = \begin{bmatrix} v_{1}\\ v_{2}\\ \vdots\\ v_{n} \end{bmatrix} = \begin{pmatrix} v_{1}\\ v_{2}\\ \vdots\\ v_{n} \end{pmatrix}$

#### Unit vector notation

A rectangular vector in $\mathbb{R}^3$ (or fewer dimensions) can be specified as the sum of the scalar multiples of the components of the vector with the members of the standard basis in $\mathbb{R}^3$. The basis is represented with the unit vectors $\hat{i} = (1, 0, 0)$, $\hat{j} = (0, 1, 0)$, and $\hat{k} = (0, 0, 1)$.

A three-dimensional vector $\vec{v}$ can be specified in the following form, using unit vector notation:

$\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$

Where $v_x$, $v_y$, and $v_z$ are the scalar components of $\vec{v}$.

## Vector basics

### Vector equivalence

Two vectors are considered equivalent if they have the same magnitude and direction, even if they they do not coincide.

### Zero vector (null vector)

A zero vector (null vector) is defined as a vector in space that has a magnitude equal to $0$ and an undefined direction.

### Position vector

A position vector is a vector that represents the position of a point in space relative to an arbitrary reference origin. If $A(x_1, y_2)$ and $B(x_1, y_2)$ are points in a coordinate plane, the unique position vector (standard component form of a vector) of vector $AB$ is given by

$AB = (x_2 - x_1, y_2 - y_1)$

### Vector magnitude

A vector’s magnitude $\lVert \vec{v} \rVert$ is the distance between its initial point and its terminal point. The magnitude of a vector in any dimension can be found using the distance formula. For example the magnitude of vector $\vec{v}$ in $\mathbb{R}^2$ is given by the formula

$\lVert \vec{v} \rVert = \sqrt{{v_1}^2 + {v_2}^2}$

The magnitude of a vector is always a positive number.

### Unit vector

A unit vector (denoted by a hat symbol $\enspace \hat{} \enspace$) is a vector with length $1$.

The unit vector components of vector $\vec{v} = \langle v_1, v_2 \rangle$ is given by

$\hat{v} = \frac{\vec{v}}{\lVert \vec{v} \rVert} = \langle \frac{v_1}{\lVert \vec{v} \rVert}, \frac{v_2}{\lVert \vec{v} \rVert} \rangle$

#### Finding a direction vector

The unit vector $\vec{u}$ in the same direction as $\vec{v}$ is given by normalizing vector $\vec{v}$.

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

The vector $\vec{a}$ of length $k$ with the same direction as $\vec{v}$ is given by multiplying $\vec{v}$’s unit vector by the scalar $k$.

$\vec{a} = k\hat{v}$

## Vector operations

For $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$:

Adding two or more vectors using the addition operation results in a new vector that is equal to the sum of the vectors.

$\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2)$

Subtracting two or more vectors using the subtraction operation results in a new vector that is equal to the difference of the vectors.

$\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2)$

### Scalar multiplication

Multiplying a vector by a scalar changes its magnitude and, if the scalar is negative, changes its direction. Multiply each vector component by the scalar.

For $\vec{v} = (v_1, v_2)$ and scalar $a$

$a \cdot \vec{v} = (a \cdot v_1, a \cdot v_2)$

### Dot (scalar) product

The dot product of two vectors results in a scalar quantity. The dot product is the sum of the products of the corresponding components.

$\vec{a} \cdot \vec{b} = \langle a_1, a_2 \rangle \cdot \langle b_1, b_2 \rangle = a_1 b_1 + a_2 b_2$

#### Angle between two vectors

The geometric interpretation of dot product states that if $\theta$ is an angle between the two nonzero vectors $a$ and $b$, where $0\degree \leq \theta \leq 180\degree$

$a \cdot b = \lVert a \rVert \lVert b \rVert \cos \theta \\ \rightarrow \cos \theta = \frac{a \cdot b}{\lVert a \rVert \lVert b \rVert} \\ \rightarrow \theta = \cos^{-1} \left( \frac{a \cdot b}{\lVert a \rVert \lVert b \rVert} \right)$
##### Determining the type of angle between two vectors
• If the dot product is zero, the two vectors are said to be orthogonal.
• If the dot product is positive, the angle between the two vectors is acute.
• If the dot product is negative, the angle between the two vectors is obtuse.
##### Determining if vectors are orthogonal, parallel, or neither

We can use the dot product to test vectors to see whether they’re orthogonal, and then if they’re not, testing to see whether they’re parallel.

1. Take the dot product of two vectors
2. If the dot product is zero, then the vectors are orthogonal
3. If the dot product is not zero, then extract the common factors of both vectors
4. If the vectors are equal, then the vectors are parallel
5. If the vectors are unequal, then the vectors are neither parallel nor orthogonal

#### Projections

Let $v$ and $w$ be two vectors in $\mathbb{R}^2$ drawn so that they have a common initial point. If we drop a perpendicular from the head of $v$ to the line determined by $w$, we determine a vector called the vector projection of $v$ onto $w$. The scalar projection of $v$ onto $w$ (also called the component of $v$ along $w$) is the length of the vector projection.

##### Vector projection

The vector projection of $v$ onto $w$

$\text{proj}_{w}v = \left( \frac{v \cdot w}{ \lVert w \rVert^2 } \right) w$
##### Scalar projection

The scalar projection of $v$ onto $w$

$\text{comp}_{w}v = \frac{v \cdot w}{ \lVert w \rVert }$

#### Work

Work equals $F \cdot d$ where $F$ is a force vector and $d$ is a displacement vector.

### Cross (vector) product

The cross product of two vectors results in a vector quantity that is orthogonal to both given vectors.

#### Properties of the cross product

It is…

• Distributive: $A \times (B + C) = A \times B + A \times C$
• Non-commutative: $A \times B \neq B \times A$
• Anti-commutative: $A \times B = -(B \times A)$
• Associative: $(A \times B) \times C = A \times (B \times C)$
• Linear: $A \times (kB) = k(A \times B)$
• Zero when vectors are parallel or anti-parallel: $A \times B = 0$ if $A$ and $B$ are parallel or anti-parallel.

It has…

• Magnitude: $\lVert A \times B \rVert = \lVert A \rVert \lVert B \rVert \sin(\theta)$ where $\theta$ is the angle between the vectors $A$ and $B$
• Direction: The direction of $A \times B$ is perpendicular to the plane formed by $A$ and $B$, and the direction of the cross product is determined by the “right-hand rule”

#### Compute cross product using matrix notation

The vector terms can be obtained by using a determinant.

$a \times b = \begin{vmatrix} i & j & k\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix}$

Cofactor expansion along the first row gives the components of the resulting vector directly.

$a \times b = \begin{vmatrix} a_2 & a_3\\ b_2 & b_3 \end{vmatrix}i - \begin{vmatrix} a_1 & a_3\\ b_1 & b_3 \end{vmatrix}j + \begin{vmatrix} a_1 & a_2\\ b_1 & b_2 \end{vmatrix}k \\[16pt] = (a_2 b_3 - a_3 b_2)i - (a_1 b_3 - a_3 b_1)j + (a_1 b_2 - a_2 b_1)k$

#### Geometric interpretation of the cross product

The cross product result is a vector that is orthogonal to both $a$ and $b$. We can use the dot product to verify that the cross product is orthogonal to both original vectors: If $u \cdot (u \times v) = 0$ and $v \cdot (u \times v) = 0$ then the cross product is orthogonal.

We can use the right hand rule to determine the direction of the orthogonal vector. Taking $i$, $j$, $k$ as the unit vector along $x$, $y$, $z$-direction.

$\sin(0) = 0$

• $i \times i = 0$
• $j \times j = 0$
• $k \times k = 0$

$\sin(90) = 1$

• $i \times j = k$
• $j \times k = i$
• $k \times i = j$

$\sin(-90) = -1$

• $j \times i = -k$
• $k \times j = i$
• $i \times k = j$

#### Magnitude of a cross product

The magnitude of the cross product of two vectors $\lVert v \times w \rVert$ is equal to the area of the parallelogram having $v$ and $w$ as adjacent sides.

#### Scalar triple product

The cross product can also be used to compute the volume of a parallelepiped in $\mathbb{R}^3$. The scalar triple product is the product of three vectors.

$| (u \times v) \cdot w |$

## Converting between vector components and magnitude & direction

### Vector magnitude from components

The magnitude of $\vec{a} = (x, y)$ is

$\lVert a \rVert = \sqrt{{a_x}^2 + {a_y}^2}$

### Vector direction from components

The direction of angle $\vec{a} = (x, y)$ is given by

$\tan\theta = \frac{a_y}{a_x} \hspace{1em} \therefore \hspace{1em} \theta = \tan^{-1} \frac{a_y}{a_x}$

Calculators solve for angles between $-90\degree$ and $90\degree$, so positive angles beyond the first quadrant must be adjusted for.

1 $\theta$
2 $\theta + 180$
3 $\theta + 180$
4 $\theta + 360$

### Vector components from magnitude & direction

The components of a vector $\vec{a}$ with magnitude $\lVert a \rVert$ and direction $\theta$ is derived by computing the unit vector of the given direction angle $(\cos\theta, \sin\theta)$ using the definition of sine and cosine given $a_x$ is adjacent and $a_y$ is opposite the angle and scaling each by the given magnitude.

$a_x = \lVert a \rVert \cos\theta \\ a_y = \lVert a \rVert \sin\theta$

Warning: This association $(\lVert a \rVert \cos\theta, \lVert a \rVert \sin\theta)$ is true only because we measured the angle $\theta$ with respect to the $x$ axis, so do not memorize these equations. Think about which side of the triangle containing the components is adjacent to the angle and which side is opposite and then assign the cosine and sine accordingly.

Lastly, adjust the positive and negative signs according to which direction each component points.

### Adding vectors in magnitude & direction form

Convert the vector given in magnitude and direction form to vector component form and perform vector addition. The summed vector’s magnitude and direction can be obtained from the resulting component form.