Page 187 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 187

178                                  VECTORS                               [CHAP. 7


                                                Supplementary Problems


                     VECTOR ALGEBRA
                     7.46.  Given any two vectors A and B, illustrate geometrically the equality 4A þ 3ðB   AÞ¼ A þ 3B.

                     7.47.  A man travels 25 miles northeast, 15 miles due east, and 10 miles due south. By using an appropriate scale,
                          determine graphically (a) how far and (b)in what direction he is from his starting position. Is it possible
                          to determine the answer analytically?  Ans.  33.6 miles, 13.28 north of east.

                     7.48.  If A and B are any two non-zero vectors which do not have the same direction, prove that mA þ nB is a
                          vector lying in the plane determined by A and B.

                     7.49.  If A, B, and C are non-coplanar vectors (vectors which do not all lie in the same plane) and
                          x 1 A þ y 1 B þ z 1 C ¼ x 2 A þ y 2 B þ z 2 C,prove that necessarily x 1 ¼ x 2 ; y 1 ¼ y 2 ; z 1 ¼ z 2 .

                     7.50.  Let ABCD be any quadrilateral and points P; Q; R; and S the midpoints of successive sides. Prove (a)that
                          PQRS is a parallelogram and  (b)that the perimeter of PQRS is equal to the sum of the lengths of the
                          diagonals of ABCD.
                     7.51.  Prove that the medians of a triangle intersect at a point which is a trisection point of each median.

                     7.52.  Find a unit vector in the direction of the resultant of vectors A ¼ 2i   j þ k, B ¼ i þ j þ 2k, C ¼ 3i   2j þ 4k.
                                          p ffiffiffiffiffi
                          Ans. ð6i   2j þ 7kÞ= 89
                     THE DOT OR SCALAR PRODUCT
                     7.53.  Evaluate jðA þ BÞ  ðA   BÞj if A ¼ 2i   3j þ 5k and B ¼ 3i þ j   2k.  Ans.  24

                     7.54.  Verify the consistency of the law of cosines for a triangle. [Hint: Take the sides of A; B; C where C ¼ A   B.
                          Then use C   C ¼ðA   BÞ ðA   BÞ.]

                     7.55.  Find a so that 2i   3j þ 5k and 3i þ aj   2k are perpendicular.  Ans.  a ¼ 4=3

                     7.56.  If A ¼ 2i þ j þ k; B ¼ i   2j þ 2k and C ¼ 3i   4j þ 2k, find the projection of A þ C in the direction of B.
                          Ans. 17/3

                     7.57.  A triangle has vertices at Að2; 3; 1Þ; Bð 1; 1; 2Þ; Cð1;  2; 3Þ.Find(a)the length of the median drawn from
                          B to side AC and  (b)the acute angle which this median makes with side BC.
                          Ans.(a)  1  p ffiffiffiffiffi  ðbÞ cos  1  p ffiffiffiffiffi
                                    26;
                                                 91=14
                                  2
                     7.58.  Prove that the diagonals of a rhombus are perpendicular to each other.
                     7.59.  Prove that the vector ðAB þ BAÞ=ðA þ BÞ represents the bisector of the angle between A and B.

                     THE CROSS OR VECTOR PRODUCT
                                                                                 p
                                                                                  ffiffiffi
                     7.60.  If A ¼ 2i   j þ k and B ¼ i þ 2j   3k, find jð2A þ BÞ ðA   2BÞj:  Ans.  5 3
                     7.61.  Find a unit vector perpendicular to the plane of the vectors A ¼ 3i   2j þ 4k and B ¼ i þ j   2k.
                                       p ffiffiffi
                          Ans.  ð2j þ kÞ= 5
                     7.62.  If A   B ¼ A   C, does B ¼ C necessarily?

                     7.63.  Find the area of the triangle with vertices ð2;  3; 1Þ; ð1;  1; 2Þ; ð 1; 2; 3Þ.  Ans.  1  p ffiffiffi 3
                                                                                    2
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