91Ó°ÊÓ

Coordinate geometry is a branch of mathematics that uses a coordinate system to define and represent geometric figures, enabling algebraic analysis. In this system, positions on a plane are given using pairs of numbers, or coordinates. For example, point \(P\) is represented by the pair \((3,2)\), and point \(Q\) by \((3,-2)\). These coordinates tell us how far the point lies from the origin along the x-axis and the y-axis.

When dealing with vectors in coordinate geometry, a vector can be thought of as an arrow pointing from one point to another. The vector \(\mathbf{a}\) from point \(P\) to point \(Q\) is calculated by taking the differences in their respective coordinates. This means you subtract the x-coordinate of point \(P\) from that of \(Q\), and likewise for the y-coordinate. This gives the direction and length of the vector, often described as its components.

Unit vectors play a crucial role in vector representation by providing direction. They are vectors with a magnitude of 1. In the Cartesian coordinate system, the most common unit vectors are \(\mathbf{i}\) and \(\mathbf{j}\), also known as the standard basis vectors.

\(\mathbf{i}\) is the unit vector in the horizontal x-direction and is represented as \((1,0)\), because it moves one unit along the x-axis and zero units along the y-axis. On the other hand, \(\mathbf{j}\) is the unit vector in the vertical y-direction represented by \((0,1)\). By using these two vectors, any vector in the plane can be easily expressed as a linear combination of \(\mathbf{i}\) and \(\mathbf{j}\). This is particularly useful for describing vectors like \(\mathbf{a}\), where knowing its horizontal and vertical components allows us to express it mathematically in a clearer, standardized form. For the given vector from \(P\) to \(Q\), we saw that it aligned with the y-axis, hence it used only \(\mathbf{j}\) for its representation.

A vector's components refer to the projections of the vector along the axes of the coordinate system. These components break down a vector into its essential parts: the x-component and the y-component.

For vector \(\mathbf{a}\) that points from \(P\) to \(Q\), its x-component \(a_x\) is calculated as \(x_Q - x_P\) and its y-component \(a_y\) as \(y_Q - y_P\). In our exercise, this resulted in \(a_x = 0\) and \(a_y = -4\).

• This component form allows us to express the vector using \(\mathbf{i}\) and \(\mathbf{j}\): \(\mathbf{a} = a_x\mathbf{i} + a_y\mathbf{j}\).
• Since \(a_x = 0\), the vector doesn’t move left or right along the x-axis, only up or down along the y-axis, determined by \(a_y\).

Expressing vectors in terms of \(\mathbf{i}\) and \(\mathbf{j}\) provides a clear visualization of their direction and magnitude, simplifying many calculations in physics and engineering. For instance, if we know how a force vector is resolved into components, we can easily determine its effect in different directions.