In Exercises 13–20, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (-8, 6), P₂ = (-2, 3)

Verified step by step guidance

1. Identify the coordinates of the initial point \( P_1 = (-8, 6) \) and the terminal point \( P_2 = (-2, 3) \).
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: \( \Delta x = -2 - (-8) \).
4. Calculate the change in the y-coordinate: \( \Delta y = 3 - 6 \).
5. Express the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) using the changes in coordinates: \( \mathbf{v} = \Delta x \mathbf{i} + \Delta y \mathbf{j} \).

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. The vector from point P₁ to point P₂ is calculated by subtracting the coordinates of P₁ from those of P₂.

Unit Vectors i and j

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

Vector Components

The components of a vector are the projections of the vector along the axes of the coordinate system. For a vector v from P₁ to P₂, the components can be found by determining the change in the x-coordinates and the change in the y-coordinates. This allows the vector to be expressed in terms of i and j, facilitating easier calculations and visualizations.