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 25

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 = (2, -1), P₂ = (5, -3)

Verified step by step guidance

Identify the coordinates of the initial point \( P_1 = (2, -1) \) and the terminal point \( P_2 = (5, -3) \).

To find the vector \( \mathbf{v} \) from \( P_1 \) to \( P_2 \), subtract the coordinates of \( P_1 \) from \( P_2 \).

Calculate the change in the x-coordinate: \( 5 - 2 \).

Calculate the change in the y-coordinate: \( -3 - (-1) \).

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

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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 expressed as the difference between their coordinates.

Unit Vectors i and j

In a Cartesian coordinate system, the unit vectors i and j represent the directions of the x-axis and y-axis, respectively. The vector i is typically represented as (1, 0), indicating movement along the x-axis, while j is represented as (0, 1), indicating movement along the y-axis. Any vector in the plane can be expressed as a linear combination of these unit vectors.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which can be visualized as moving from one point to another in the coordinate plane. For vectors represented by points P₁ and P₂, the vector v from P₁ to P₂ is calculated by subtracting the coordinates of P₁ from those of P₂. This operation yields a new vector that describes the direction and distance from P₁ to P₂.