Friday, 12 September 2014

VECTOR ADDITION PROBLEMS AND SOLUTIONS


Problem1 : What is (1, 4, - 3) + 2(0, 1, 5) ?
(1, 6, 7)
Problem 2: If a creature was initially sitting at the point (3,4) in the plane, and then moved to the point (-2,5), what is the vector which describes the creature's displacement?
If v is the displacement vector for the creature, we have that (3, 4) + v = (- 2, 5) . Solving for v yields v = (- 5, 1) .
Problem 3: Express the vector (2, 2, 5) in terms of unit vectors.
(2, 2, 5) = 2i +2j +5k .
Problem4 : Consider two vectors u and v in the plane which comprise two sides of a triangle. What vector corresponds to the third side?
Either u - v or v - u are acceptable answers. (The difference will be whether the arrow for the third side points towards u or towards v ).
Problem5 : Find the sum of the vectors which make up the vertices of a regular pentagon centered at the origin.
The sum of the vectors, which you can find using the geometric method for vector addition, should turn out to be zero. An easy way to see this is by noticing that if you rotate the pentagon by an angle of 2Π/5 , you will recover exactly the same same pentagon, with exactly the same vectors defining its vertices. Thus, whatever vector you obtain by adding up the five vectors, it should remain unchanged under such a rotation. Only the zero-vector (the origin itself) doesn't move when you rotate the plane, hence this is the only possible candidate for the result of the sum.