Vector Diagrams: Summary

Here’s a reminder of the key points you should know about 3D vectors.

We can use vectors to describe movements in 2D space…

… but also movements in 3D space.

Just like 2D vectors, 3D vectors have a magnitude… and a direction.

And we can give vectors names.

We can represent them using point to point notation… [BA]

… or column vector notation by adding a third value… [(2 5 3) and then pulse the 3]

… or unit vector notation by adding this third unit vector, k. [2i + 5j + 3k and then pulse the 3k.]

To describe the position of a point, we can use coordinates… 

Or we can use a position vector instead.

Next, adding 3D vectors works just like adding 2D vectors.

Here’s what it looks like when we add two vectors in column vector notation.

And here’s what it looks like in unit vector form.

Here’s what it looks like when we multiply by a scalar in column vector notation…

… and in unit vector form.

If two vectors are multiples of one another, that means they’re parallel.

Next, we can find the magnitude of a 3D vector using 3D pythagoras.

And to find the distance between two points, we can either use this formula…

Or take the position vector for each point…

… subtract one from the other…

… and then find the magnitude of that new vector.

Finally, if you’re given a vector and asked to find the unit vector that points in the same direction…

Divide the original vector…by its magnitude.