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Scalar Multiplication of Vectors

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Scalar Multiplication of Vectors

Study Guide

Key Definition

To multiply a vector by a scalar, multiply each component by the scalar: $n \vec{u} = \langle n u_1, n u_2 \rangle$

Important Notes

The magnitude of a scaled vector is $|c| \cdot |\vec{v}|$
If $n$ is negative, the direction reverses
Associative Property: $c(d\vec{u}) = (cd)\vec{u}$
Distributive Property: $(c+d)\vec{u} = c\vec{u} + d\vec{u}$
Identity Property: $1 \cdot \vec{u} = \vec{u}$

Mathematical Notation

$\vec{u}$ denotes a vector

$|\vec{u}|$ denotes the magnitude of $\vec{u}$

$\langle u_1, u_2 \rangle$ represents a vector with components

$+$ represents addition

Remember to use proper notation when solving problems

Why It Works

The scalar multiplication scales the magnitude by $|n|$ and keeps the direction unchanged unless $n$ is negative.

Remember

Multiply each component of the vector by the scalar: $n \vec{u} = \langle n u_1, n u_2 \rangle$

Quick Reference

Associative Property:$c(d\vec{u}) = (cd)\vec{u}$

Distributive Property:$(c+d)\vec{u} = c\vec{u} + d\vec{u}$

Understanding Scalar Multiplication of Vectors

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $n \vec{u} = \langle n u_1, n u_2 \rangle$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $3 \times \langle 2, 5 \rangle$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

If a skateboarder moves along the vector $\langle -1, 3 \rangle$ and doubles the speed, find the new vector.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens to the direction of $\langle 4, -3 \rangle$ when multiplied by $-2$?

Challenge Quiz

Single Choice Quiz

Advanced

Find $-3 \times \langle 1, -2 \rangle$

Recap

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Review key concepts and takeaways