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.

Textbook | NCERT |

Class | Class 12th |

Subject | Maths |

Chapter | 25 |

Exercise | 25.1 |

Category | RD Sharma Solutions |

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

**Question 1: Ten cards numbered 1 through 10 are placed in a box, mixed **

### Question 1. If and , find

**Solution:**

Given, and .

=> =

=> =

=> =

=> =

=> =

Now,

=> =

=> =

=> = √91

### Question 2(i). If and , find the value of

**Solution:**

Given, and

=> =

=> =

=> =

=> =

=> =

Now,

=> =

=> =

=> =

### Question 2(ii). If and , find the magnitude of

**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,

=> =

### Question 4. Find the magnitude of vector

**Solution:**

Given

=>

=> =

=> =

=> =

=> =

Unit vector is,

=> =

=> =

=> = √74

### Question 5. If and , then find

**Solution:**

Given, and

=> =

=> =

=> =

=> =

=> =

=> =

=> =

=> =

Now,

=> =

=> =

=> =

**Question 6. If ****and ****, find **** **

**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 , findSolution:

We know that,

=>

=>

=>

=>

=>

Also,

=>

And

=>

=>

=>

=>

=>

=>

Question 14. Find the angle between 2 vectors and , ifSolution:

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. If, and , find the angle between andSolution:

Given that,, and

We know that,

=>

=>

=>

=>

=>

=>

=>

=>

Question 17. What inference can you draw if andSolution:

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 thatSolution:

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 thatSolution:

We know that,

=>

=>

=>

=>

=>

=>

=>

=>

=>

Hence proved.

Question 24. Define and prove that , where is the angle between andSolution:

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. If, and , findSolution:

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 andSolution:

Given that is perpendicular to both and .

=> ……….(1)

=> ……….(2)

Also,

=> …….(3)

Let

From eq(1),

=> d_{1}+ 4d_{2}+ 2d_{3}= 0

From eq(2),

=> 3d_{1}– 2d_{2}+ 7d_{3}= 0

From eq(3),

=> 2d_{1}– d_{2}+ 4d_{3}= 15

On solving the 3 equations we get,

d_{1 }= 160/3, d_{2 }= -5/3, and d_{3 }= -70/3,

=>

Question 28. Find a unit vector perpendicular to each of the vectorsand, whereand.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.

By triangle law of addition,

=>

=>

=>

=>

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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