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Magnitude and Direction of Vectors

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Magnitude and Direction of Vectors

Study Guide

Key Definition

The magnitude of a vector $\overrightarrow{PQ}$ is the distance between points $P$ and $Q$. It is calculated using <a href="distance-formula">$|\overrightarrow{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$</a>. The direction is the angle $\theta$ with the horizontal: first compute $\tan(\theta) = \frac{y_2 - y_1}{x_2 - x_1}$, then $\theta = \arctan\bigl(\frac{y_2 - y_1}{x_2 - x_1}\bigr)$, adjusting for the correct quadrant (e.g., using $\mathrm{atan2}$).

Important Notes

Use the Distance Formula to find vector magnitude: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Direction of a vector is measured in degrees or radians (default degrees; specify units clearly)
The angle $\theta$ is calculated using $\theta = \arctan\bigl(\frac{y_2 - y_1}{x_2 - x_1}\bigr)$, adjusted for the correct quadrant (e.g., via atan2)
Vectors can be represented graphically as arrows
Magnitude is always a positive number

Mathematical Notation

$\overrightarrow{PQ}$ represents a vector from $P$ to $Q$

$|\overrightarrow{PQ}|$ denotes the magnitude of vector $\overrightarrow{PQ}$

$\tan(\theta)$ is the tangent of angle $\theta$

$\sqrt{}$ denotes the square root

$\theta$ is the angle the vector makes with the horizontal axis

Remember to use proper notation when solving problems

Why It Works

Vectors have both magnitude and direction, which are essential for representing quantities in physics and engineering. The mathematical formulas help quantify these properties.

Remember

The magnitude of a vector is found using distance calculations, and direction is found through trigonometric functions like $\tan(\theta)$.

Quick Reference

Magnitude of Vector:<a href="distance-formula">$|\overrightarrow{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$</a>

Direction of Vector:$\theta = \arctan\bigl(\frac{y_2 - y_1}{x_2 - x_1}\bigr)$ (adjust for quadrant; use atan2)

Understanding Magnitude and Direction of Vectors

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Video explanation of this concept

Beginner

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Beginner Explanation

A vector has a starting point and an endpoint. The magnitude is the distance between these two points, calculated using $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the magnitude of a vector $\overrightarrow{PQ}$ with initial point $P(1, 1)$ and endpoint $Q(5, 3)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A drone flies from point $A(2, 3)$ to point $B(5, 8)$. Calculate the direction of the flight path.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

A boat sails in a direction forming an angle of $\theta$ with the shoreline. If $\tan(\theta) = \frac{3}{4}$, what is the angle $\theta$?

Challenge Quiz

Single Choice Quiz

Advanced

Find the magnitude of vector $\overrightarrow{RS}$ where $R(-2, -3)$ and $S(4, 1)$.

Recap

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