When two vectors and are drawn from a common point, the angle between them is ???. (a) Using vector techniques, show that the magnitude of their vector sum is given by (b) If and have the same magnitude, for which value of ?? will their vector sum have the same magnitude as or ?
Solution 90P Let us prove the magnitude of the resultant of addition of two vector by using the vector addition by triangular method. Step 1 (a) The triangular law of vector addition states that: When the two sides of a rectangle represents two vector by its magnitude and direction taken in same order, the the third side of the triangle, taken in reverse order will represents the resultant of the two vector in magnitude and direction. Let us now consider the following figure. In this figure A is represented by OA and B by AB, then the resultant vector OB= R represents the resultant of the addition of A and B . The angle between the vector is . Now let us extend the line OA and drop a perpendicular BC on the line. So OCB makes a right angle triangle. Then the length of side OB is given by ..........................(1) Now let us consider the triangle ABC. From the trigonometry we can write that Similarly we can write that Now we can also write that Now putting the value of OC and BC in equation (1) we can write that Now putting the value and and we can write that (proved) (b) The condition of the problem is , i.e. all the sides of the vectors should be equal. Which means this three vector will form a equilateral triangle. And from the geometry we know that, all angle of an equilateral triangle is 60°. Hence the angle between the vector and is 60°.