> See also: > - [Dot products and duality | Chapter 9, Essence of linear algebra](https://www.youtube.com/watch?v=LyGKycYT2v0) > - [2D Vector Calculator](https://www.xitalogy.com/apps/math/vec2dcalc/) # Vectors There are multiple ways to view **vectors** that vary between different disciplines (physics, computer science, mathematics). Generally speaking, a vector can be thought of as a set of instructions used to get from one point to another. - In a basic 2D environment, a vector would only have 2 dimensions, one for the change in the x-value and another for the change in the y-value. The **magnitude** of a vector is it's length. ## Properties of Vectors ## Vector Components A vector with >1 componenet can be “resolved”, or broken down, into its separate components. ## Vectors There are multiple ways to view **vectors** that vary between different disciplines (physics, computer science, mathematics). Generally speaking, a vector can be thought of as a set of instructions used to get from one point to another. The **magnitude** of a vector is it's length. The direction of a vector is specified by two elements: --- To vectors are *equal* when they have the *same magnitude AND direction*. Their position within the field does not need to be the same. ## Vector Components Just like a point can be described as a pair of coordinates $(x,y)$ in a [[Coordinate Systems|Cartesian coordinate system]], a vector $\vec{A}$ in a plane can be described by a collection of numbers called **components**. ![[small.svg]] The number of components that a vector contains is called its *dimensions*. --- > [!summary]- Magnitude and Orientation from Vector Components > > When given the components of a vector (i.e. $\langle A_x, A_y \rangle$), the magnitude and the orientation of the vector are found by: > > $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$ > $\theta = \arctan (\frac{A_y}{A_x})$ > > The orientation of the vector, $\theta$, is the angle between the vector and the positive direction of the $x$-axis. #### Addition Using Vector Components ![[Pasted image 20250110115921.png|225]] ### Unit Vectors A **unit vector** is a dimensionless vector with a magnitude of 1 that can be used to *store the direction of a vector*. Unit vectors are represented using a ^ symbol: - $\hat{i} \equiv$ (1, positive $x$-direction) - $\hat{j} \equiv$ (1, positive $y$-direction) - $\hat{k} \equiv$ (1, positive $z$-direction) $\vec{A} = A_x \hat{i} + A_y \hat{j}$ ## Vector Operations ### Dot Products While matrix multiplication/products is performed between two matrices, a **dot product** is defined as the *product of two vectors*. Dot product ### Vector Point Duality Vectors can often be confused with points, so they are often represented vertically within brackets, similar to the format of [[Matrices]]. $\begin{bmatrix} 2 \\ 2 \end{bmatrix}$