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


P₁ = (-1, 6), P₂ = (7, -5)

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 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 along the x-axis and y-axis, respectively. The vector i is typically represented as (1, 0), while j is represented as (0, 1). When expressing a vector in terms of i and j, we decompose it into its horizontal and vertical components, allowing for a clearer understanding of its direction and magnitude.

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. For a vector v from P₁ to P₂, the components can be found by calculating the difference in the x-coordinates and the y-coordinates of the points. This results in a vector expressed as v = (x-component)i + (y-component)j, which simplifies the analysis of the vector's direction and length.