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

1. Identify the coordinates of the initial point \( P_1 = (2, -1) \) and the terminal point \( P_2 = (5, -3) \).
2. To find the vector \( \mathbf{v} \) from \( P_1 \) to \( P_2 \), subtract the coordinates of \( P_1 \) from \( P_2 \).
3. Calculate the change in the x-coordinate: \( 5 - 2 \).
4. Calculate the change in the y-coordinate: \( -3 - (-1) \).
5. Express the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) using the changes in the x and y coordinates.

Key Concepts

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.