notes/education/math/MATH1060 (trig)/Vectors.md

A vector is a mathematical concept that denotes direction and magnitude. They're often notated using an arrow (\vec{v}), or with a bold, lowercase letter. (v).

Vectors are often denoted as a matrix with two rows: \begin{bmatrix}1 \\2\end{bmatrix}

Component Form

If \vec{v} is a vector with the initial point (x_y,\ y_i), and a terminal point (x_t,\ y_t), we can express \vec{v} in component form as \vec{v} = \langle x_t - x_i,\ y_t, -y_i \rangle

Magnitude

The magnitude of a vector is |\vec{v}| = \sqrt{a^2 + b^2}

Direction

The direction of a vector is \theta = \tan^-1(\frac{b}{a}).

Addition

To find \vec{u} + \vec{v}, we can put one vector on the end of another vector. The resulting vector will share the same tail as the first vector, and the same head as the second vector.

Scalar Multiplication

A scalar is just a real number. Scalar multiplication is multiplying a vector with a real number. This will scale or shrink a vector, but does not change the direction it points at.

We do not multiply two vectors together.

Unit Vector

A vector with a magnitude of 1 is a unit vector.

If \vec{v} is a nonzero vector, the unit vector can be found using the equation \vec{u} = \dfrac{1}{|\vec{v}|}\vec{v} . In other words, to find a unit vector, divide the vector by its magnitude.

i, j Notation

Every 2d vector has a horizontal component and a vertical component. The horizontal unit vector could be written as i = \langle 1, 0 \rangle, and the vertical unit vector could be written as j = \langle 0, 1 \rangle Every vector can be made up using a combination of these standard unit vectors.

Trigonometric Form

Given a vector \vec{v} with a magnitude |\vec{v}| and direction \theta:

The component form is given as:

 \vec{v} = \langle \cos \theta,\ |\vec{v}|\sin\theta \rangle 

Standard position

• A vector is in standard position if the initial point is at (0, 0).

The Dot Product

The dot product of two vectors \vec{u} = \langle a, b \rangle and \vec{v} = \langle c, d \rangle is \vec{u} * \vec{v} = ac + bd.

• Given that \vec{u} = \langle -7, 3 \rangle, and \vec{v} = \langle -3, 4 \rangle, find \vec{u} * \vec{v}.
• \vec{u} * \vec{v} = -7 * -4 + 3 * 4