Master

Components of a Vector

Master components of a vector with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Components of a Vector

Study Guide

Key Definition

In a two-dimensional coordinate system, any vector can be broken into x-component and y-component. The vector in the component form is $\vec{v} = \langle v_x , v_y \rangle$

Important Notes

The vector and its components form a right angled triangle.
Magnitude of the vector is $|v| = \sqrt{v_x^2 + v_y^2}$
To find direction of the vector, solve $\tan(\theta) = \frac{v_y}{v_x}$ for $\theta$, and use the signs of v_x and v_y to determine the correct quadrant.

Mathematical Notation

$\vec{v}$ represents a vector

$v_x$ and $v_y$ are the x and y components of the vector respectively

$\theta$ is the angle between the vector and its x-component

Remember to use proper notation when solving problems

Why It Works

The decomposition of a vector into its components is based on the Pythagorean Theorem and trigonometry. This allows us to describe a vector in terms of its magnitude and direction.

Remember

The x and y components of a vector $\vec{v}$ can be given as $v_x = v\cos(\theta)$ and $v_y = v\sin(\theta)$ respectively.

Quick Reference

Magnitude:$|v| = \sqrt{v_x^2 + v_y^2}$

Direction:$\tan(\theta) = \frac{v_y}{v_x}$ (check quadrant based on signs of v_x and v_y)

Understanding Components of a Vector

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A vector can be broken down into its x and y components. This is done using basic trigonometry.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Given the vector $\vec{v} = \langle 3 , 4 \rangle$, what are the x and y components?

Real-World Problem

Question Exercise

Intermediate

Rocket Launch

A rocket is launched and is represented by the vector $\vec{v} = \langle 3 , 4 \rangle$. Calculate the magnitude of the vector.

Advanced Challenge

Thinking Exercise

Advanced

Think About This

Given the vector $\vec{v} = \langle 1 , 1 \rangle$, find the direction of the vector in degrees.

Challenge Quiz

Single Choice Quiz

Advanced

Given the magnitude of a vector is 5 and the angle it makes with the x-axis is 30 degrees, what are the x and y components?

Recap

Watch & Learn

Review key concepts and takeaways