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Working with vectors

Working with vectors

Last Update on June 12, 2023

1 mins

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

Vectors are a geometric object with a magnitude (length) and direction. They are represented by an arrow, where the arrow shows the direction and the length represents the magnitude.

So looking at the diagram we can see that vector u vector has a greater magnitude than v vector. Vectors can also be described in terms of the points they pass between. So

\{\begin{cases} \vec{u} = \vec{P Q} \\ \vec{v} = \vec{P S} \end{cases}

with the arrow over the top showing the direction.

You can use vectors as a geometric algebra, expressing other vectors in terms of u vector and v vector. For example

This may seem slightly counter-intuitive at first. But if we add in some possible figures you can see how it works. If $\vec{u}$ moves 5 units to the left and $\vec{v}$ moves 1 unit to the right (−left) and 3 units down.

Then $\vec{P R} = \vec{u} + \vec{v} = 5$ units to the left −1 unit to the right and 3 units down = 4 units to the left and 3 units down.