A Note There are two perpendicular directions to any plane. Thus, another unit vector perpendicular to will be ˆ But that will be a consequence of Example Find the area of a triangle having the points A( , , ), B( , , ) and C( , , ) as its vertices. Solution We have . The area of the given triangle is Now, = − Therefore + = Thus, the required area is Example Find the area of a parallelogram whose adjacent sides are given by the vectors Solution The area of a parallelogram with as its adjacent sides is given by Now Therefore + + and hence, the required area is .
EXERCISE . . Find . Find a unit vector perpendicular to each of the vector , where .
If a unit vector makes angles with , with π π and an acute angle θ with ˆ k , then find θ and hence, the components of . . Show that . Find λ and µ if .
Given that . What can you conclude about the vectors ? . Let the vectors be given as a i a j a k b i b j b k c i c j c k .
Then show that . If either then . Is the converse true? Justify your answer with an example.
. Find the area of the triangle with vertices A( , , ), B( , , ) and C( , , ). . Find the area of the parallelogram whose adjacent sides are determined by the vectors .
Let the vectors be such that , then is a unit vector, if the angle between is (A) π / (B) π / (C) π / (D) π / . Area of a rectangle having vertices A, B, C and D with position vectors – , k i – , respectively is (A) (B) (C) (D) Miscellaneous Examples Example Write all the unit vectors in XY-plane. Solution Let be a