Dear Uncle Colin

Do you have any tips for sketching three-dimensional vectors? Every time we have an A-level question, my teacher says “draw a diagram!” but I don’t know how to draw in 3D.

- Got A Useless Sketching Situation

Hi, GAUSS, and thank you for your message!

Three-dimensional vectors often seem to catch people out at A-level - everyone is so used to doing everything in 2D, adding an extra dimension appears to make everything harder.

However.

As far as I can remember, there are no 3D situations (at A-level) where you need to be able to sketch in 3D: a 2D sketch is always good enough. More to the point, your sketch doesn’t need to be especially accurate, it just needs to be big enough that you can mark on it all of the details you know:

• the position vectors of any points (especially points of intersection)
• the angle between any pair of lines
• any other distances that interest you

A typical question that requires a sketch will involve finding a point on a line that’s as close as possible to another point, or the minimum distance to a line. Both of these, once you draw the picture, really only require right-angled trigonometry: either you can determine the distance of your desired point from the crossing point and figure out how many direction-vectors you need to move along to get there, or you can find the perpendicular distance you need directly.

If you’re looking for the area of a shape, on the other hand, you will need to pay attention to where lines are parallel, and possibly think about how to divide your shape up into smaller bits, the areas of which you can find.

The key take-away, though, is don’t get hung up about an accurate sketch. Big and well-labelled is better than accurate. Sketches are for your benefit, not anyone else’s!

Hope that helps,

- Uncle Colin