WORKSHEET ON VECTOR ALGEBRA USING DOT PRODUCT

Problem 1 :

Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

Solution

Problem 2 :

Prove by vector method that the median to the base of an isosceles triangle is perpendicular to the base.

Solution

Problem 3 :

Prove by vector method that an angle in a semi-circle is a right angle.

Solution

Problem 4 :

Prove by vector method that the diagonals of a rhombus bisect each other at right angles.

Solution

Problem 5 :

Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle.

Solution

Problem 6 :

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is

Solution

Problem 7 :

Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area

Solution

Problem 8 :

If G is the centroid of a ∆ABC , prove that

Area of triangle GAB = Area of triangle GBC = Area of triangle GCA = (1/3) Area of triangle ABC

Solution

Problem 9 :

Using vector method, prove that

cos (α - β) = cos α cos β + sin α sin β

Solution

Problem 10 :

Prove by vector method that

sin (α + β) = sin α cos β + cos α sin β

Solution

Problem 11 :

A particle acted on by constant forces

is displaced from the point (1, 2, 3) to the point (5, 4, 1) . Find the total work done by the forces

Solution

Problem 12 :

Forces of magnitudes 5√2 and 10√2 units acting in the directions

respectively act on a particle which is displaced from the point with position vector 

to the point with position vector 

Find the work done by the forces.

Solution

Problem 13 :

Find the magnitude of the direction cosines of torque represented by 

about the position vector 

acting through the position vector is 

Solution

Problem 14 :

Find the torque of the resultant of the three forces represented by

acting at the point with position vector

about the point with position vector

Solution