Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

8. Vectors

Unit Vectors and i & j Notation

Trigonometry

8. Vectors

Unit Vectors and i & j Notation

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2:44 minutes

Problem 4.33

Textbook Question

The magnitude and direction angle of v are ||v|| = 12 and θ = 60°. Express v in terms of i and j.

Verified step by step guidance

Step 1: Understand that the vector $ \mathbf{v} $ can be expressed in terms of its magnitude and direction angle using the formula $ \mathbf{v} = ||\mathbf{v}|| (\cos \theta \mathbf{i} + \sin \theta \mathbf{j}) $.

Step 2: Substitute the given magnitude $ ||\mathbf{v}|| = 12 $ and direction angle $ \theta = 60^\circ $ into the formula.

Step 3: Calculate $ \cos 60^\circ $ and $ \sin 60^\circ $. Recall that $ \cos 60^\circ = \frac{1}{2} $ and $ \sin 60^\circ = \frac{\sqrt{3}}{2} $.

Step 4: Substitute $ \cos 60^\circ $ and $ \sin 60^\circ $ into the expression for $ \mathbf{v} $: $ \mathbf{v} = 12 \left( \frac{1}{2} \mathbf{i} + \frac{\sqrt{3}}{2} \mathbf{j} \right) $.

Step 5: Distribute the magnitude 12 into the expression: $ \mathbf{v} = 12 \times \frac{1}{2} \mathbf{i} + 12 \times \frac{\sqrt{3}}{2} \mathbf{j} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnitude of a Vector

The magnitude of a vector, denoted as ||v||, represents its length in a given space. In this case, ||v|| = 12 indicates that the vector has a length of 12 units. Understanding magnitude is crucial for visualizing the size of the vector and is often the first step in vector analysis.

Direction Angle

The direction angle θ of a vector indicates its orientation relative to a reference axis, typically the positive x-axis. Here, θ = 60° means the vector is positioned at a 60-degree angle from the x-axis. This angle is essential for decomposing the vector into its components along the x and y axes.

Vector Components

Vector components are the projections of a vector along the coordinate axes, usually represented as i (x-axis) and j (y-axis). To express vector v in terms of i and j, we use the formulas v_x = ||v|| * cos(θ) and v_y = ||v|| * sin(θ). For θ = 60°, this results in specific values for the x and y components, allowing for a complete representation of the vector.