# Vector Product (Part-5) | S N Dey | Class 12

In the previous article, we discussed Short Answer Type Questions of Vector Product. In this article, we will discuss the solutions of Long Answer Type Questions (1-11) of Ex-2A in the chapter Product of Two Vectors as given in the Chhaya Publication Book of aforementioned chapter of S N De book.

### Vector Product | S N Dey mathematics class 12 Solutions of Ex-2A

Applying vectors, show that

Solution.

let

By vector method show that,

an angle inscribed in a semi- circle is a right angle.

Solution.

let be the inscribed angle .

To prove :

is the diameter and is the center of the semi-circle.

Now, by the law of triangle of vectors, we get

Hence, by and we get,

###### the parallelogram whose diagonals are equal is a rectangle.

Solution.

let be a parallelogram and

Hence, is a parallelogram.

###### the perpendicular bisectors of the sides of a triangle are concurrent.

Solution.

let be the perpendicular bisectors of the sides respectively, where be the point of intersection of

Suppose that position vectors of with respect to are respectively.

Let be the midpoint of and so the position vectors of are respectively.

We have to prove that

Hence, from and we get,

Hence, the perpendicular bisectors of the sides of a triangle are concurrent.

medium to the base of an isoscales triangle is perpendicular to the base.

Solution.

let is an isoscales triangle where

Suppose that the position vectors of with respect to are respectively, being the origin.

let be the mid point of the base so that

Also, by we get,

Hence follows the result.

are three given points. Find the angle between the vectors

Solution.

By the question, the position vectors of are given by

If is the angle between the vectors

Three vectors are such that if and then find the value of

Solution.

The scalar product of the vector with the unit vector along the sum of vectors is equal to one. Find the value of

Solution.

Now, the unit vector along the sum of vectors is given by :

By question,

Let and be three given vectors ; If and are perpendicular to each other , find

Solution.

If and are perpendicular to each other , then

Let and Find a vector which is perpendicular to both

Solution.

We know that denotes a vector which is perpendicular to both and

By the question and using we can say that

where is any scalar, so that

If and are the position vectors of points and respectively, then find the angle between and Deduce that and are collinear.

Solution.

If be the angle between and then

Hence, we can deduce that and are collinear.

Express the vector as sum of two vectors such that one is parallel to the vector and the other is perpendicular to

Solution.

Let where is parallel to and

Since is parallel to so that , where is a non-zero scalar.

Since

Hence, by we get by putting the value of

###### If then show by an example that the converse of this statement is not always true.

Solution.

Let

Here, clearly but

Hence by the result follows.

If and find the vector which is perpendicular to both and and which satisfies the relation

Solution.

let

By question,

From and we get by cross multiplication,

Let be the position vectors of the vertices of a triangle ; prove that the area of the triangle is

Solution.

Area of is given by :

Given If and is perpendicular to and find in component form the vector

Solution.

We first compute the value of

If is perpendicular to and then

Hence, by we get