Direction, Distance and Displacement

Walk fifty metres east and then thirty metres west, and the thirty undoes part of the fifty: you finish twenty metres east of where you started. But you did walk eighty metres to get there. Those are two different quantities and the paper asks for both. How far did she walk adds the legs together; how far is she from home cancels the opposite ones. Reading the last line of the question is the only way to know which is wanted, so read it before you start adding anything.

With a longer journey, split it into two separate accounts: everything east or west, and everything north or south. They never interact, so you get two small sums rather than one confusing one. Here thirty east against fifty west leaves twenty west. Fifty north against seventy south leaves twenty south. So the walker finishes twenty metres west and twenty metres south of the start. Keeping the two axes apart on paper — literally two little columns — is what stops the mistakes.

Once you have the net movement, the way home is simply that net reversed: swap each direction for its opposite and keep the distances the same. Finishing twenty metres west and twenty metres south means home is twenty metres east and twenty metres north. You do not have to retrace the original path in reverse — only the net matters, which is why this is quick. The wrong options are built from individual legs, offering you fifty metres or seventy metres, so work out the net first and then flip it.

Coordinates are the same idea in different clothing. They are written across first and then up, so from the origin, three to the right and two up lands you on three comma two. East adds to the first number and west subtracts from it; north adds to the second and south subtracts. That is exactly the cancelling from the earlier steps, just written as arithmetic instead of drawn as arrows, and it means a journey question and a coordinate question can be solved with the same two columns.

Turning questions add one more layer. A right turn always moves you one step clockwise, wherever you started: facing north you end up facing east, and facing south you end up facing west. Four right turns bring you back to where you began, so five right turns is the same as one and six is the same as two — divide by four and keep the remainder. And remember that on your left depends on the direction you are travelling, not on which side of the page something is drawn. Turn the page so the arrow points up, and then your own left is the correct one. Quiz and exercises now.