Expert: Scotto - 10/11/2008

Question
Hello. I'm having a little difficulty with doing three of these math questions and since it's the school holiday's I can't exactly ask my math teacher. If you could explain what steps I need in order to get the answers, that would be great.

1. A boy trying to salvage a log from a river, flowing with speed 5 k.p.h. pulls the rope he has tied to the log at 3 k.p.h. at right angles to the bank of the river. The log is sinking at the rate of 0.2 k.p.h. If the rope is pulled at an angle of 10 degrees to the horizontal, find the resultant speed of the log.

2. Find the cosines of angles alpha, beta and gamma between the vector 'a'=12i-16j+21k and the positive x, y, z axes respectively. These Cosines are called the direction cosines of vector 'a'.
Then find the value of the sum of the squares of direction cosines.

3. A vector 'b' acts at 60 degrees to the x axis and at 45 degrees to the y axis. At what angle does this vector act with respect to the z axis?

Thank you.
Ming.

Answer
In my first response, I only answer question 1.  I then looked back and remembered there were two more questions.

2. The cos() is given by the near side over the hypoteneuse.  The near side is 12, and 16, and 21 for x, y, and z, respectively.
The length of the hypoteneus is √(12²+16²+21²).  This will give you the cosine of each of the angles.  Note that the sum of squares of these cosines makes the top look just like the bottom, except for the fact that bottom has a √ involved.

3. Note that the sum of squares of cos() in each direction is 1.  The angle with the z azis must be the √[1-angle(x)²-angle(y)²].