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

Geometric Vectors

Trigonometry

8. Vectors

Geometric Vectors

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2:57 minutes

Problem 41

Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.v = 3i - 4j

Verified step by step guidance

insert step 1: Understand that a unit vector is a vector with a magnitude of 1 that points in the same direction as the given vector.

insert step 2: Calculate the magnitude of the vector \( \mathbf{v} = 3\mathbf{i} - 4\mathbf{j} \) using the formula \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the components of the vector.

insert step 3: Substitute the components of the vector into the magnitude formula: \( \|\mathbf{v}\| = \sqrt{3^2 + (-4)^2} \).

insert step 4: Simplify the expression under the square root to find the magnitude of the vector.

insert step 5: Divide each component of the vector \( \mathbf{v} \) by its magnitude to find the unit vector: \( \mathbf{u} = \left( \frac{3}{\|\mathbf{v}\|} \right)\mathbf{i} + \left( \frac{-4}{\|\mathbf{v}\|} \right)\mathbf{j} \).

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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 in space, calculated using the formula √(x² + y²) for a two-dimensional vector (x, y). In the context of the vector v = 3i - 4j, the magnitude is √(3² + (-4)²) = √(9 + 16) = √25 = 5. This value is essential for normalizing the vector to find the unit vector.

Direction of a Vector

The direction of a vector is determined by the angle it makes with a reference axis, typically the x-axis. In the case of the vector v = 3i - 4j, the direction can be visualized in the Cartesian plane, where the vector points from the origin to the point (3, -4). Understanding direction is crucial when finding a unit vector, as it ensures the unit vector points in the same way as the original vector.