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Adding and Subtracting Vectors

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Adding and Subtracting Vectors

Study Guide

Key Definition

A vector has both direction and magnitude(size). You can add and subtract vectors by adding or subtracting their corresponding components.

Important Notes

Vectors are usually represented in the form $\langle u_1, u_2 \rangle$
Vector addition and subtraction operates component-wise.

Mathematical Notation

$\langle u_1 + v_1 , u_2 + v_2 \rangle$ denotes vector addition

$\langle u_1 - v_1 , u_2 - v_2 \rangle$ denotes vector subtraction

Remember to use proper notation when solving problems

Why It Works

The rules for adding and subtracting vectors work because they are derived from the geometric properties of vectors. The magnitude and direction of the resultant vector are determined by the mathematical properties of the original vectors.

Remember

Always add or subtract corresponding components of the vectors.

Quick Reference

Vector Addition:$u + v = \langle u_1 + v_1 , u_2 + v_2 \rangle$

Vector Subtraction:$u - v = \langle u_1 - v_1 , u_2 - v_2 \rangle$

Understanding Adding and Subtracting Vectors

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Add or subtract the corresponding components of the vectors.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Find $u + v$ if $u = \langle 3, 4 \rangle$ and $v = \langle 5, -1 \rangle$.

Real-World Problem

Question Exercise

Intermediate

Airplane Navigation

An airplane is flying due east with a speed of 300 km/h. There is a wind blowing due north at 50 km/h. If the plane wants to maintain its eastward direction, how should the pilot adjust the plane's course?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $u = \langle 3, 4 \rangle$ and $v = \langle 5, -1 \rangle$, what is the vector $u - v$?

Challenge Quiz

Single Choice Quiz

Advanced

Given vectors $u = \langle 3, 4 \rangle$, $v = \langle 5, -1 \rangle$, and $w = \langle -1, 2 \rangle$, find $u + v - w$.

Recap

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