VECTOR ALGEBRA

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 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 Solution Let is a scalar, i.e., Now ( ) ( −λ + λ Now, since β is to be perpendicular to α r , we should have .

i.e., ( ) ( −λ − + λ = or λ = Therefore = Miscellaneous Exercise on Chapter . Write down a unit vector in XY-plane, making an angle of ° with the positive direction of x -axis. . Find the scalar components and magnitude of the vector joining the points P( x , y , z ) and Q( x , y , z ).

. A girl walks km towards west, then she walks km in a direction ° east of north and stops. Determine the girl’s displacement from her initial point of departure. .

If , then is it true that ? Justify your answer. . Find the value of x for which x i is a unit vector.

. Find a vector of magnitude units, and parallel to the resultant of the vectors . If , find a unit vector parallel to the vector . Show that the points A( , – , – ), B( , , – ) and C( , , ) are collinear, and find the ratio in which B divides AC.

. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are externally in the ratio : . Also, show that P is the mid point of the line segment RQ. .

The two adjacent sides of a parallelogram are and Find the unit vector parallel to its diagonal. Also, find its area. . Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ±       .

Let . Find a vector which is perpendicular to both and , and . The scalar product of the vector with a unit vector along the sum of vectors λ + is equal to one. Find the value of λ .

. If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to . Prove that , if and only if are perpendicular, given Choose the correct answer in Exercises to . .

If θ is the angle between two vectors , then only when (A) π < θ < (B) π ≤θ ≤ (C) < θ < π (D) ≤ θ ≤ π . Let be two unit vectors and θ is the angle between them. Then is a unit vector if (A) π θ = (B) π θ = (C) π θ = (D) π θ = . The value of ˆ ˆ .( is (A) (B) – (C) (D) .

If θ is the angle between any two vectors , then when θ is equal to (A) (B) π (C) π (D) π Summary