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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2:09 minutes

Problem 26

Textbook Question

In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (-3, 0), P₂ = (-2, -2)

Verified step by step guidance

Identify the coordinates of the initial point \( P_1 = (-3, 0) \) and the terminal point \( P_2 = (-2, -2) \).

Calculate the change in the x-coordinate: \( \Delta x = x_2 - x_1 = -2 - (-3) \).

Calculate the change in the y-coordinate: \( \Delta y = y_2 - y_1 = -2 - 0 \).

Express the vector \( \mathbf{v} \) in terms of \( i \) and \( j \) using the changes in coordinates: \( \mathbf{v} = \Delta x \cdot \mathbf{i} + \Delta y \cdot \mathbf{j} \).

Substitute the calculated \( \Delta x \) and \( \Delta y \) into the expression for \( \mathbf{v} \).

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Key Concepts

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

Vectors

A vector is a mathematical object that has both magnitude and direction. In a two-dimensional space, a vector can be represented as an ordered pair of coordinates, indicating its position relative to a reference point. For example, the vector from point P₁ to P₂ can be calculated by subtracting the coordinates of P₁ from those of P₂.

Unit Vectors i and j

In a Cartesian coordinate system, the unit vectors i and j represent the directions along the x-axis and y-axis, respectively. The vector i is typically represented as (1, 0), while j is represented as (0, 1). When expressing a vector in terms of i and j, we decompose it into its horizontal and vertical components, allowing for a clearer understanding of its direction and magnitude.

Vector Components

The components of a vector are the projections of the vector along the coordinate axes. For a vector v from P₁ to P₂, the x-component is found by subtracting the x-coordinates of P₁ from P₂, and the y-component is found similarly for the y-coordinates. This breakdown into components is essential for performing vector operations and understanding the vector's behavior in a plane.