Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.


P₁ = (2, -5), P₂ = (-6, 6)

Verified step by step guidance

1. Identify the coordinates of the initial point \( P_1 = (2, -5) \) and the terminal point \( P_2 = (-6, 6) \).
2. To find the vector \( \mathbf{v} \) from \( P_1 \) to \( P_2 \), calculate the difference in the x-coordinates: \( x_2 - x_1 = -6 - 2 \).
3. Calculate the difference in the y-coordinates: \( y_2 - y_1 = 6 - (-5) \).
4. Express the vector \( \mathbf{v} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) using the differences calculated: \( \mathbf{v} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} \).
5. Substitute the calculated differences into the expression for \( \mathbf{v} \) to write it in terms of \( \mathbf{i} \) and \( \mathbf{j} \).

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. 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 along the x-axis and y-axis, respectively. The vector i is typically represented as (1, 0), while j is represented as (0, 1). Any vector in the plane can be expressed as a linear combination of these unit vectors, allowing for a clear representation of its components.

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 points P₁ and P₂, the vector v can be calculated by subtracting the coordinates of P₁ from those of P₂. This operation yields a new vector that indicates the direction and distance from P₁ to P₂.