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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2:33 minutes

Problem 5

Textbook Question

In Exercises 5–12, sketch each vector as a position vector and find its magnitude.v = 3i + j

Verified step by step guidance

Step 1: Understand the vector notation. The vector \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \) is given in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) represents the unit vector in the x-direction and \( \mathbf{j} \) represents the unit vector in the y-direction.

Step 2: Sketch the vector. To sketch \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \), start at the origin (0,0) on a coordinate plane. Move 3 units in the positive x-direction and 1 unit in the positive y-direction. Draw an arrow from the origin to the point (3,1). This arrow represents the vector \( \mathbf{v} \).

Step 3: Use the Pythagorean theorem to find the magnitude of the vector. The magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) is given by \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \).

Step 4: Substitute the components of the vector into the magnitude formula. For \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \), substitute \( a = 3 \) and \( b = 1 \) into the formula: \( \|\mathbf{v}\| = \sqrt{3^2 + 1^2} \).

Step 5: Simplify the expression under the square root. Calculate \( 3^2 \) and \( 1^2 \), then add the results to find the expression under the square root.

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Key Concepts

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

Position Vectors

A position vector represents a point in space relative to an origin. In a two-dimensional Cartesian coordinate system, a position vector can be expressed in terms of its components along the x-axis and y-axis, typically denoted as 'i' and 'j'. For example, the vector v = 3i + j indicates a point located 3 units along the x-axis and 1 unit along the y-axis.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector expressed as v = ai + bj, the magnitude is given by the formula |v| = √(a² + b²). This quantifies how far the vector extends from the origin to its endpoint in the coordinate system.

Vector Components

Vector components break down a vector into its individual parts along the coordinate axes. In the vector v = 3i + j, the component '3' represents the horizontal (x-axis) contribution, while '1' represents the vertical (y-axis) contribution. Understanding these components is essential for visualizing the vector and calculating its magnitude.