unit vector in XY-plane (Fig . ). Then, from the figure, we have x = cos θ and y = sin θ (since | | = ). So, we may write the vector as cos sin θ + θ ...
( ) Clearly, | | = cos sin θ + θ = Fig . Also, as θ varies from to π , the point P (Fig . ) traces the circle x + y = counterclockwise, and this covers all possible directions. So, ( ) gives every unit vector in the XY-plane.
Example If , , and k are the position vectors of points A, B, C and D respectively, then find the angle between Deduce that are collinear. Solution Note that if θ is the angle between AB and CD, then θ is also the angle between Now = Position vector of B – Position vector of A ( ) = + Therefore | | = ( ) ( ) ( ) + − Similarly and |CD | uuur Thus cos θ = = ( ) ( ) ( )( ) ( )( ) + − = − Since ≤ θ ≤ π , it follows that θ = π . This shows that are collinear. Alternatively , which implies that are collinear vectors.
Example Let be three vectors such that each one of them being perpendicular to the sum of the other two, find Solution Given Now = + + = Therefore Example Three vectors satisfy the condition . Evaluate the quantity Solution Since , we have = or = Therefore ... ( ) Again, = or ... ( ) Similarly = – .
... ( ) Adding ( ), ( ) and ( ), we have = – or µ = – , i.e., µ = Example If with reference to the right handed system of mutually perpendicular unit vectors , then express in the form is parallel to is perpendicular to