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

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.

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.

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

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

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

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

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 $\vec{v}$.

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

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

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

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

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.

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

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

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 $k$.

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

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.

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

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 $a$

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

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

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

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

$a \cdot b = \lVert a \rVert \lVert b \rVert \cos \theta
\\
\rightarrow \cos \theta = \frac{a \cdot b}{\lVert a \rVert \lVert b \rVert}
\\
\rightarrow \theta = \cos^{-1} \left( \frac{a \cdot b}{\lVert a \rVert \lVert b \rVert} \right)$

- 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**.

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.

- Take the dot product of two vectors
- If the dot product is
**zero**, then the vectors are**orthogonal** - If the dot product is
**not zero**, then extract the common factors of both vectors - If the vectors are equal, then the vectors are
**parallel** - If the vectors are unequal, then the vectors are
**neither parallel nor orthogonal**

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.

The vector projection of $v$ onto $w$

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

The scalar projection of $v$ onto $w$

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

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

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

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”

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.

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

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.

$\sin(0) = 0$

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

$\sin(90) = 1$

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

$\sin(-90) = -1$

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

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.

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.

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

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

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

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.

Quadrant | Adjustment |
---|---|

1 | $\theta$ |

2 | $\theta + 180$ |

3 | $\theta + 180$ |

4 | $\theta + 360$ |

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.

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.