If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.

Verified step by step guidance

1. Identify the coordinates of the points: \( P_1 = (-2, 3) \) and \( P_2 = (-1, 5) \).
2. Calculate the change in the x-coordinates: \( \Delta x = x_2 - x_1 = -1 - (-2) \).
3. Calculate the change in the y-coordinates: \( \Delta y = y_2 - y_1 = 5 - 3 \).
4. Express the vector \( \mathbf{v} \) in terms of \( i \) and \( j \) using the changes in coordinates: \( \mathbf{v} = \Delta x \cdot \mathbf{i} + \Delta y \cdot \mathbf{j} \).
5. Substitute the calculated values of \( \Delta x \) and \( \Delta y \) into the expression for \( \mathbf{v} \).

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 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 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 can be expressed as a linear combination of these unit vectors.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which results in a new vector. For 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 not only gives the direction from P₁ to P₂ but also provides the components of the vector in terms of i and j.