In the first unit on vectors, we were introduced to quantities having both magnitude and direction.

We considered their geometric representation as a directed line segment or arrow.

In This Unit

• We will introduce vectors in a Cartesian coordinate system and consider the convenience that this new model provides.
• We will refer to these as algebraic vectors otherwise called vectors in component form.

Vector \(\vec{v}\) is defined to have magnitude \(10\) and a direction of north \(45^\circ\) east. This vector, \(\vec{v}\), is an example of a geometric vector.

This was a typical representation of the concept throughout the first vector unit.

Question: Can we represent the same vector, \(\vec{v}\), in a different way?

Answer: Yes, by using the notion of algebraic vectors.

To represent a geometric vector algebraically, position the tail of the vector, \(\vec{v}\), at the origin of a Cartesian \((x,y)\)-plane.

In our case, \(\lvert \vec{v}\rvert=10\) and the direction is north \(45^\circ\) east, which corresponds to an angle of \(45^\circ\) CCW (counter-clockwise) from the positive \(x\)-axis (this is defined as standard position).

We can now represent vector \(\vec{v}\) using coordinates \(x\) and \(y\), where \((x,y)\) is the location of the tip of vector \(\vec{v}\) when its tail is positioned at the origin.

This raises the question: How do we find the coordinates \(x\) and \(y\)?

Therefore, vector \(\vec{v}\) can be written algebraically as an ordered pair \((5\sqrt{2},5\sqrt{2})\).

The ordered pair \((5\sqrt{2},5\sqrt{2})\) is an example of an algebraic vector.

In the ordered pair, \(x=5\sqrt{2}\) and \(y=5\sqrt{2}\) are referred to as the \(x\) and \(y\) components of the vector \(\vec{v}\). As a result, \(\vec{v}=(5\sqrt{2},5\sqrt{2})\) may be said to be in component form.