Expert: Josh - 8/18/2006

Question
Hi Paul !

I am about to begin highschool, unfortunately I do not know how to preform rotations (clockwise and counterclockwise) on a cartesian plane using a protractor, ruler and and compass. I was not able to grasp this during my grade eight year and would like to know how before the school year starts.I have tried to talk to my parents and many other people but it seemed they didn't study this  area when they were in school. I also searched many websites but found that what they discussed was too advanced for me. Could you please, tell how to      
preform rotations on a cartesian plane, and recommend any websites that could help me?

Thanks

Mandy

Answer
Hi Mandy,

I think your question was addressed to Paul, but I'll answer it anyhow. In the following discussion, I will refer to the diagram at http://www.kidport.com/Grade6/Math/MeasureGeo/MeasuringAngles.htm

Please make sure you have it ready, perhaps in a printed form when you go through the steps below.

With reference to the diagram on the webpage (see link above), let's suppose that we are interested in rotating the vector BA by some angle, "w".

Note: a vector is just a line segment -- in this case, extending from point B to point A -- and it has a direction (i.e., angle) and magnitude (i.e., length) associated with it.

By convention, we usually place the protractor as shown in the diagram, with the flat side parallel to the bottom edge of a page. Accordingly, angular measurements are taken from the east in the counter-clockwise direction.

So, if a vector (i.e., line segment with a fixed length and direction) points to the right hand side, it has an angle of 0 degree by definition. If we rotate this by 45 degrees (implicitly in the anti-clockwise direction), the new vector will point in the North East direction. If the vector were rotated by 90 degrees from its starting position (initially at zero degree), it would point to the North. Whenever two vectors are at 90 degrees with each other, we say that they are perpendicular.

Now, looking at the diagram again, suppose that we want to measure the direction of vector BA. We place the protractor with the plus sign (+) centered at point B. Make sure that the horizontal line on the protractor coincides with BC. Now, we take the reading along the line which aligns with the vector BA. It should read 70 degrees.

Next, supppose that we are interested in rotating the vector BA by an arbitrary angle "w". Let "r" be the magnitude of vector BA (just measure the distance from point B to A). With a compass, you set it a distance of "r" apart. Now plant one of its legs at point B. You should be able to draw a circular arc which passes through point A. If you complete one revolution, this procedure will produce a circle with radius, "r".

Terminology: On the protractor (see diagram), the line going from 0 degree to 180 degrees is referred to as the "baseline".

If we are asked to rotate BA by 20 degrees (anti-clockwise), then, we place the protractor in such a way that the plus sign (+) is positioned at point B, and the baseline of the protractor aligns with vector BA.
i.e., the line BA now coincides with 0 degree.

Now, along the protractor, find the angular grid corresponding to 20. Using a pencil, mark this position. For reference, we refer to this 20 degree position on the arc as P. Now, remove the protractor. Using a ruler, joining point B to point P, you form a vector BP obtained by rotating BA by 20 degrees. (Aside: since BA was oriented at 70 degrees, and the rotation added 20 degrees to it, BP is now at 90 degrees with respect to the x-y cartesian plane; meaning that it points up north)

If we were asked to rotate BA by -30 degrees (the minus is interpreted as clockwise, rather than anti-clockwise, rotation), we can instead align vector BA with the 90 degree marking on the protractor. Of course, the (+) sign still coincides with point B. Then, we proceed in the clockwise direction, counting back 10, 20, 30 degrees. Where it says 60 on the protractor, we mark this on paper and call this, say, point R. Remove the protractor and join B to R. This is effectively BA rotated by -30 degrees.

I hope you get the idea.

Josh