Introduction to Vectors

Study Notes

Introduction to Vectors

Vectors are mathematical objects with both magnitude (size) and direction.
They are often represented graphically as arrows, where the arrow's length represents the magnitude and direction represents the vector's direction.
Vectors represent physical quantities like displacement, velocity, and force, all having magnitude and direction.
Vectors can be added, subtracted, and scaled.

Components of a Vector

Vectors can be broken down into components along different axes.
A 2D vector can be decomposed into horizontal (x) and vertical (y) components.
Vector components are denoted like $\vec{A} = A_x \hat{i} + A_y \hat{j}$.
Unit vectors $\hat{i}$ and $\hat{j}$ represent the x-axis and y-axis directions respectively.

Vector Addition and Subtraction

Vectors add/subtract using the parallelogram law or component-wise addition/subtraction.
The resultant vector represents the sum or difference of the original vectors.
To add, place one vector's tail at the other's head; the resultant points from the first vector's tail to the second's head.
To subtract vector $\vec{B}$ from $\vec{A}$, add the negative of $\vec{B}$ to $\vec{A}$.

Scaling Vectors

Multiplying a vector by a scalar changes its magnitude.
Positive scalars maintain the vector's direction, negative scalars reverse the direction.

Vector Magnitude

The magnitude (length) of vector $\vec{A}$ is denoted by $||\vec{A}||$ or $A$.
For a 2D vector $\vec{A} = A_x \hat{i} + A_y \hat{j}$, magnitude is calculated as $A = \sqrt{A_x^2 + A_y^2}$.

Unit Vectors

A unit vector has a magnitude of 1.
Unit vectors indicate direction without specifying magnitude, denoted with a 'hat' (e.g., $\hat{i}$, $\hat{j}$, $\hat{k}$).

Applications of Vectors

Vectors are crucial in physics (displacement, velocity, acceleration, force), engineering (forces, moments), and computer graphics (movement, transformations).
They concisely represent and manipulate quantities with both magnitude and direction.
Calculating distance between points involves finding the displacement vector's magnitude.
Describing motion uses vectors for velocity and acceleration.

Vector Dot Product

The dot product of two vectors is a scalar.
For vectors $\vec{A}$ and $\vec{B}$, the dot product is written as $\vec{A} \cdot \vec{B}$.
Geometrically, it's the product of magnitudes and the cosine of the angle between them.
Mathematically, it's calculated by summing the products of corresponding components: $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$ (3D vectors).

Vector Cross Product

The cross product of two vectors is a vector.
It's represented as $\vec{A} \times \vec{B}$ for vectors $\vec{A}$ and $\vec{B}$.
Calculated in 3D space, it produces a vector perpendicular to both input vectors.
Magnitude is the product of magnitudes and the sine of the angle between them.
Only defined in three-dimensional space.

Vector Spaces

A vector space is a collection of all possible vectors with specific vector addition and scalar multiplication rules.
These spaces are vital for understanding vector properties and behaviors in abstract contexts.
Examples include Euclidean and infinite-dimensional spaces, depending on vector dimensions and operations.

Description

This quiz covers the fundamentals of vectors, including their definitions, components, and operations. Learn how vectors are utilized in various contexts, such as physics, to describe physical quantities with both magnitude and direction. Test your understanding of vector addition, subtraction, and component resolution.