91Ó°ÊÓ

A position vector represents the position of a point in space relative to the origin. In our case, the vector \( \mathbf{a} = \langle -4, -7 \rangle \) is a position vector in two-dimensional space. This vector has two components: an \(x\)-component of \(-4\) and a \(y\)-component of \(-7\). These components tell us how far and in which direction to move from the origin \((0, 0)\) to reach the point \((-4, -7)\).

When sketching a position vector in a coordinate plane, like \( \mathbf{a} \):

• Start at the origin \((0, 0)\).
• Move horizontally by \(-4\) units (since \(x = -4\)).
• Then, move vertically by \(-7\) units (since \(y = -7\)).
• Draw an arrow pointing from the origin to the location \((-4, -7)\) to represent the vector.

Position vectors are important because they describe a precise location in space using minimal information.

The magnitude of a vector refers to its length or size. It's a measure of how far the vector extends from the origin to its pointed end. To find the magnitude of a vector \( \mathbf{a} = \langle x, y \rangle \), apply the Pythagorean theorem:\[\| \mathbf{a} \| = \sqrt{x^2 + y^2}\]
In our example, the vector \( \mathbf{a} \) has components \(x = -4\) and \(y = -7\). Substituting these into the formula gives:\[\| \mathbf{a} \| = \sqrt{(-4)^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65}\]
This \( \sqrt{65} \) is the magnitude of vector \( \mathbf{a} \), and it represents how long the vector is. Magnitudes are always positive or zero; they are like distances. For practical use, you may approximate \( \sqrt{65} \approx 8.06 \) if needed.

Two-dimensional space can be visualized like a flat sheet or a plane. It has two directions: horizontal (or \(x\)-axis) and vertical (or \(y\)-axis). Vectors in this space, like \( \mathbf{a} = \langle -4, -7 \rangle \), have two components which are aligned along these axes.

Understanding two-dimensional space is crucial because:

• Most basic geometry problems are solved in this plane.
• It's foundational for visualizing graphs, shapes, and vectors.
• Understanding it makes transitioning to three dimensions or more easier.

In mathematics and physics, two-dimensional space provides a simple, yet powerful way to deal with concepts like velocity, force, and position by representing them as vectors. It helps us model real-world situations that we can visualize and interpret using relatively straightforward calculations and diagrams.