WORD PROBLEMS IN VECTOR ALGEBRA

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 :

OA = OB = radii

Let O is the center of the circle, C is the midpoint of the chord. Then, AC = BC.

When the product of two vectors is equal to 0, then the those two vectors involving are perpendicular.

Problem 2 :

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

Solution :

In the triangle AOB, OA = OB because it is isosceles triangle. OC is the median of the isosceles triangle.

Problem 3 :

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

Solution :

O is the center of the circle, P is the point of the semicircle.

Since the product of AP and PB is equal to 0, then angle measure P is right angle.

Problem 4 :

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

Solution :

So, the diagonals are perpendicular.

Problem 5 :

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

Solution :

Given ABCD is a parallelogram. In parallelogram, opposite sides are equal.

Problem 6 :

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

Solution :

Area of quadrilateral

= Area of triangle ABC + Area of triangle ADC

Problem 7 :

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

Solution :

Area of parallelogram = product of adjacent sides

Area of parallelogram (ABCD) = AB x AD

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 :