# RD Sharma Class 12 Ex 25.1 Solutions Chapter 25 Vector or Cross Product

Here we provide RD Sharma Class 12 Ex 25.1 Solutions Chapter 25 Vector or Cross Product for English medium students, Which will very helpful for every student in their exams. Students can download the latest RD Sharma Class 12 Ex 31.1 Solutions Chapter 31 Probability book pdf download. Now you will get step-by-step solutions to each question.

## RD Sharma Class 12 Ex 25.1 Solutions Chapter 25 Vector or Cross Product

Solution:

Given, and  .

=>

=>

=>

=>

=>

Now,

=>

=>

=> = √91

Solution:

Given,  and

=>

=>

=>

=>

=>

Now,

=>

=>

=>

Solution:

Given,  and

=>

=>

=>

=>

=>

Now,

=>

=>

=> = √6

### Question 3(i). Find a unit vector perpendicular to both the vectors  and

Solution:

Given  and

A vector perpendicular to 2 vectors is given by

=>

=>

=>

=>

=>

Unit vector is given by

=>  =

=>

=> = 3

=> Unit vector is,

=>

### Question 3(ii). Find a unit vector perpendicular to the plane containing the vectors  and  .

Solution:

Given, and

A vector perpendicular to 2 vectors is given by

=>

=>

=>

=>

=>

Unit vector is given by

=>

=>

=>

=> Unit vector is,

=>

Solution:

Given

=>

=>

=>

=>

=>

Unit vector is,

=>

=>

=> = √74

Solution:

Given, and

=>

=>

=>

=>

=>

=>

=>

=>

Now,

=>

=>

=>

Solution:

Given,  and

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

### Question 7(i). Find a vector of magnitude 49, which is perpendicular to both the vectors  and

Solution:

Given,  and

A vector perpendicular to 2 vectors is given by

=> =

=>  =

=>  =

=>  =

=>

Magnitude of vector is given by,

=>

=>

=>

=>

=> Vector is,

### Question 7(ii). Find the vector whose length is 3 and which is perpendicular to the vector and

Solution:

Given,  and

A vector perpendicular to 2 vectors is given by

=>

=>

=>

=>

=>

Magnitude of vector is given by,

=>

=>

=>

=> = 27

=> Unit vector is,

=>

=>

Required vector is,

=>

### Question 8(i). Find the parallelogram determined by the vectors:  and

Solution:

Given that, and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus the area of parallelogram is,

=>

=>

=> Area = 6 square units.

### Question 8(ii). Find the parallelogram determined by the vectors:  and .

Solution:

Given that, and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus, the area of parallelogram is,

=>

=>

=> Area =

### Question 8(iii). Find the area of the parallelogram determined by the vectors:  and

Solution:

Given that, and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus the area of parallelogram is,

=>

=>

=> Area =

### Question 8(iv). Find the area of the parallelogram determined by the vectors:  and

Solution:

Given that,  and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus the area of parallelogram is,

=>

=>

=> Area =

### Question 9(i). Find the area of the parallelogram whose diagonals are: and

Solution:

Given, and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus the area of parallelogram is,

=>

=>

=> Area = 15/2 = 7.5 square units

### Question 9(ii). Find the area of the parallelogram whose diagonals are:  and

Solution:

Given, and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus the area of parallelogram is,

=>

=>

=> Area =

### Question 9(iii). Find the area of the parallelogram whose diagonals are:  and

Solution:

Given, and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus the area of parallelogram is,

=>

=>

=> Area =

### Question 9(iv). Find the area of the parallelogram whose diagonals are:  and

Solution:

Given, and

=> Area of the parallelogram is

=>

=>

=>

=>

=>

Thus the area of parallelogram is,

=>

=>

=> Area =

=> Area = 24.5

### Question 10. If  ,  and , compute  and  and verify these are not equal.

Solution:

Given and

=>

=>

=>

=>

=>

=>  =

=>  =

=>

=>

=>

=>

=>

=>

=>

=>

=>

=> is not equal to

=> Hence verified.

### Question 11. If ,  and , find

Solution:

We know that,

=>

=>

We know that  is 1, as  is a unit vector

=>

=>

=>

Also,

=>

And

=>

=>

=>

=>

=>

=>

### Question 12. Given , , , , , being a right-handed orthogonal system of unit vectors in space, show that ,  and  is also another system.

Solution:

To show that  and  is a right-handed orthogonal system of unit vectors, we need to prove:

(1)

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

=>

(2)

=>

=>

=>

=>

=>

(3)

=>

=>

=>

=>

=>

(4)

=>

=>

=>

=>

=>

Hence proved.

Question 13. If and , find
Solution:

We know that,
=>
=>
=>
=>
=>
Also,
=>
And
=>
=>
=>
=>
=>
=>
Question 14. Find the angle between 2 vectors and , if
Solution:
Given
=>
=>, as  is a unit vector.
=>
=>
=>
Question 15. If , then show that , where m is any scalar.
Solution:
Given that
=>
=>
=>
Using distributive property,
=>
If two vectors are parallel, then their cross-product is 0 vector.
=>  and  are parallel vectors.
=>
Hence proved.
Question 16. Ifand , find the angle between  and
Solution:
Given that,and
We know that,
=>
=>
=>
=>
=>
=>
=>
=>
Question 17. What inference can you draw if and
Solution:
Given, and
=>
=>
Either of the following conditions is true,
1.
2.
3.
4.is parallel to
=>
=>
Either of the following conditions is true,
1.
2.
3.
4. is perpendicular to
Since both these conditions are true, that implies atleast one of the following conditions is true,
1.
2.
3.
Question 18. If and are 3 unit vectors such that and . Show that ,and  form an orthogonal right handed triad of unit vectors.
Solution:
Given, and
As,
=>
=>  is perpendicular to both and .
Similarly,
=>  is perpendicular to both and
=>  is perpendicular to both and
=> and  are mutually perpendicular.
As, and  are also unit vectors,
=> and  form an orthogonal right-handed triad of unit vectors
Hence proved.
Question 19. Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B, and C are A(3, -1, 2), B(1, -1, 3), and C(4, -3, 1).
Solution:
Given A(3, -1, 2), B(1, -1, 3) and C(4, -3, 1).
Let,
=>
=>
=>
Plane ABC has two vectors  and
=>
=>
=>
=>
=>
=>
=>
=>
A vector perpendicular to both and is given by,
=>
=>
=>
=>
=>
To find the unit vector,
=>
=>
=>
=>
Question 20. If a, b and c are the lengths of sides BC, CA and AB of a triangle ABC, prove that and deduce that
Solution:
Given that and
From triangle law of vector addition, we have
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
Similarly,
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
Hence proved.
Question 21. If and , then find . Verify that and  are perpendicular to each other.
Solution:
Given, and
=>
=>
=>
=>
=>
Two vectors are perpendicular if their dot product is zero.
=>
=>
=>
=>
Hence proved.
Question 22. If  and  are unit vectors forming an angle of , find the area of the parallelogram having and  as its diagonals.
Solution:
Given and forming an angle of .
Area of a parallelogram having diagonals  and  is
=>
=>
=>
Thus area is,
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =  square units
Question 23. For any two vectors  and  , prove that
Solution:
We know that,
=>
=>
=>
=>
=>
=>
=>
=>
=>
Hence proved.
Question 24. Define and prove that , where  is the angle between and
Solution:
Definition of : Let  and be 2 non-zero, non-parallel vectors. Then , is defined as a vector with the magnitude of , and which is perpendicular to both the vectors  and .
We know that,
=>
=>
=> ……………..(eq.1)
And as,
=>
=>
Substituting in (eq.1),
=>
=>
Question 25. Ifand , find
Solution:

We know that,
=>
=>
=>
=>
=>
As ,
=>
=>
=>
=>
Thus,
=>
=>
=>
Question 26. Find the area of the triangle formed by O, A, B when
Solution:
The area of a triangle whose adjacent sides are given by  and  is
=>
=>
=>
=>
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =  square units.
Question 27. Let  and. Find a vector which is perpendicular to bothand and
Solution:
Given that  is perpendicular to both  and .
=>  ……….(1)
=>  ……….(2)
Also,
=>  …….(3)
Let
From eq(1),
=> d1 + 4d2 + 2d3 = 0
From eq(2),
=> 3d1 – 2d2 + 7d3 = 0
From eq(3),
=> 2d1 – d2 + 4d3 = 15
On solving the 3 equations we get,
d= 160/3, d= -5/3, and d= -70/3,
=>
Question 28. Find a unit vector perpendicular to each of the vectors and , where  and .
Solution:
Given that,  and
Let
=>
=>
=>
Let
=>
=>
=>
A vector perpendicular to both  and  is,
=>
=>
=>
=>
To find the unit vector,
=>
=>
=>
=>
=>
Question 29. Using vectors, find the area of the triangle with the vertices A(2, 3, 5), B(3, 5, 8), and C(2, 7, 8).
Solution:
Given, A(2, 3, 5), B(3, 5, 8), and C(2, 7, 8)
Let,
=>
=>
=>
Then,
=>
=>
=>
=>
=>
=>
=>
=>
The area of a triangle whose adjacent sides are given by  and  is
=>
=>
=>
=> Area =
=> Area =
=> Area = √61/2
Question 30. If are three vectors, find the area of the parallelogram having diagonals  and .
Solution:
Given,
Let,
=>
=>
=>
=>
=>
=>
=>
The area of the parallelogram having diagonals  and  is
=>
=>
=>
=> Area =
=> Area =
=> Area =
=> Area = √21/2
Question 31. The two adjacent sides of a parallelogram are and . Find the unit vector parallel to one of its diagonals. Also, find its area.
Solution:
Given a parallelogram ABCD and its 2 sides AB and BC.
=>
=>
=>
=>
Unit vector is,
=>
=>
=>
=>
Area of a parallelogram whose adjacent sides are given is
=>
=>
=>
Thus area is,
=> Area =
=> Area =
=> Area =
=> Area = 11 √5 square units
Question 32. If either or , then . Is the converse true? Justify with example.
Solution:
Let us take two parallel non-zero vectors  and
=>
For example,
and
=>
=>
But,
=>
=>
Hence the converse may not be true.
Question 33. If and, then verify that .
Solution:
Given, and
=>
=>
=>
=>  …..eq(1)
Now,
=>
=>
And,
=>
=>
Thus,
=>
=>  …eq(2)
Thus eq(1) = eq(2)
Hence proved.
Question 34(i). Using vectors find the area of the triangle with the vertices A(1, 1, 2), B(2, 3, 5), and C(1, 5, 5).
Solution:
Given, A(1, 1, 2), B(2, 3, 5), and C(1, 5, 5)
=>
=>
=>
Now 2 sides of the triangle are given by,
=>
=>
=>
=>
=>
=>
=>
=>
Area of the triangle whose adjacent sides are given is
=>
=>
=>
Thus area of the triangle is,
=> Area =
=> Area =
=> Area = √61/2
Question 34(ii). Using vectors find the area of the triangle with the vertices A(1, 2, 3), B(2, -1, 4), and C(4, 5, -1).
Solution:
Given, A(1, 2, 3), B(2, -1, 4), and C(4, 5, -1)
=>
=>
=>
Now 2 sides of the triangle are given by,
=>
=>
=>
=>
=>
=>
=>
=>
Area of the triangle whose adjacent sides are given is
=>
=>
=>
Thus area of the triangle is,
=> Area =
=> Area =
=> Area = √274/2
Question 35. Find all the vectors of magnitude  that are perpendicular to the plane of and .
Solution:
Given, and
A vector perpendicular to both  and  is,
=>
=>
=>
Unit vector is,
=>
=>
=>
=>
Now vectors of magnitude  are given by,
=>
=> Required vectors,
Question 36. The adjacent sides of a parallelogram are and . Find the 2 unit vectors parallel to its diagonals. Also, find its area of the parallelogram.
Solution:
Given,  and
=>
=>
=>
Unit vector is,
=>
=>
=>
Area is given by ,

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