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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2:12 minutes

Problem 4.40

Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = -5j

Verified step by step guidance

insert step 1> Identify the given vector $ \mathbf{v} = -5\mathbf{j} $. This vector is in the direction of the negative y-axis.

insert step 2> Calculate the magnitude of the vector $ \mathbf{v} $. The magnitude of a vector $ \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} $ is given by $ \|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2} $. For $ \mathbf{v} = -5\mathbf{j} $, $ a = 0, b = -5, c = 0 $.

insert step 3> Substitute the values into the magnitude formula: $ \|\mathbf{v}\| = \sqrt{0^2 + (-5)^2 + 0^2} = \sqrt{25} $.

insert step 4> Simplify the expression to find the magnitude: $ \|\mathbf{v}\| = 5 $.

insert step 5> Find the unit vector $ \mathbf{u} $ by dividing the vector $ \mathbf{v} $ by its magnitude: $ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \frac{-5\mathbf{j}}{5} = -\mathbf{j} $.

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Key Concepts

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

Unit Vector

A unit vector is a vector that has a magnitude of one and indicates direction. To find a unit vector in the same direction as a given vector, you divide the vector by its magnitude. This process normalizes the vector, allowing it to retain its direction while having a standardized length.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y² + z²) for three-dimensional vectors. For a two-dimensional vector like v = -5j, the magnitude is simply the absolute value of its components. Understanding how to compute the magnitude is essential for normalizing the vector to create a unit vector.

Direction of a Vector

The direction of a vector indicates the path along which it acts and is often represented by the angle it makes with a reference axis. In the case of the vector v = -5j, it points directly downward along the y-axis. Recognizing the direction is crucial when finding a unit vector, as it ensures that the resulting unit vector maintains the same orientation.