NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**Download NCERT solutions for Vector Algebra as PDF.**

## NCERT Solutions class 12 Maths Vector Algebra

**1. Write down a unit vector in XY-plane making an angle of **** with the positive direction of ****axis.**

**Ans. **Let be the unit vector in XY-plane such that XOP =

Therefore, OP = 1 ……….(i)

By Triangle Law of Addition of vectors,

In , =

[Unit vector along OX is and that is along OY is ][Dividing and multiplying by OP in R.H.S.]

[Using eq. (i)]

**2. Find the scalar components and magnitude of the vector joining the points P**** and Q**

**Ans. **Given points are P and Q

Position vector of point P = =

And Position vector of point Q = =

Now = Position vector of Q – Position vector of P

=

= =

Scalar components of the vector are the coefficients of in , i.e.,

And magnitude of vector =

**3. A girl walks 4 km towards west, then she walks 3 km in a direction **** east of north and stops. Determine the girl’s displacement from her initial point of departure.**

**Ans. **Let the initial point of departure is origin (0, 0) and the girl walks a distance OA = 4 km towards west.

Through the point A, draw a line AQ parallel to a line OP, which is East of North, i.e., in East-North quadrant making an angle of with North.

Again, let the girl walks a distance AB = 3 km along this direction

= ……….(i) [ Vector is along OX’]

Now, draw BM perpendicular to axis.

In by Triangle Law of Addition of vectors,

Dividing and multiplying by AB in R.H.S.,

=

……….(ii)

Girl’s displacement from her initial point O of departure to final point B,

= =

**4. If **** then is it true that **** Justify your answer.**

**Ans. **Given:

Either the vectors are collinear or form the sides of a triangle.

Case I: Vectors are collinear.

Let and

Then

Also, = AC = AB + BC =

Case II: Vectors form a triangle.

Here also by Triangle Law of vectors,

But [ Each side of a triangle is less than sum of the other two sides]

is true only when vectors and are collinear vectors.

**5. Find the value of **** for which **** is a unit vector.**

**Ans. **Since is a unit vector,

Therefore,

Squaring both sides,

**6. Find a vector of magnitude 5 units and parallel to the resultant of the vectors **** and **** **

**Ans. **Given: Vectors and

Let vector be the resultant vector of and

= +

=

Required vector pf magnitude 5 units and parallel (or collinear) to resultant vector is

=

=

=

=

**7. If **** and **** find a unit vector parallel to the vector **** **

**Ans. **Given: Vectors and

Let

=

=

=

A unit vector parallel to the vector is

=

=

**8. Show that the points A**** B**** and C (11, 3, 7) are collinear and find the ratio in which B divides AC.**

**Ans. **Given: Points A B and C (11, 3, 7).

Position vector of point A =

Position vector of point B =

Position vector of point C =

Now = Position vector of point B – Position vector of point A

= = =

Again = Position vector of point C – Position vector of point B

= = =

Again = Position vector of point C – Position vector of point A

= = =

Now = =

Therefore, points A, B, C are either collinear or are the vertices of a triangle ABC.

Again AB + BC = = AC

**Now to find ratio in which B divides AC**

Let the point B divides AC in the ratio

Therefore, using section formula, Position vector of point B is

=

Comparing coefficients of both sides, we get

Therefore, required ratio = = : 1 = 2 : 3

**9. Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are **** and **** externally in the ratio 1 : 2. Also, show that P is the middle point of line segment RQ.**

**Ans. **Since position vector of point R dividing the join of P and Q externally in the ratio 1 : 2 = is given by

Again position vector of the middle point of the line segment RQ

= (Position vector of point R – Position vector of point Q)

= = = = Position vector of point P (given)

Therefore, P is the middle point of the line segment RQ.

**10. Two adjacent sides of a parallelogram are **** and **** Find the unit vector parallel to its diagonal. Also, find its area.**

**Ans. **Let ABCD is a parallelogram.

Given: The vectors representing two adjacent sides of this parallelogram say,

and

Now vectors along the diagonals and of the parallelogram are

and

= =

And = =

Therefore, Unit vectors parallel to (or along) diagonals are

and

and and

and

Now Area of parallelogram = =

= =

= = = sq. units

**11. Show that the direction cosines of a vector equally inclined **** to the axes OX, OY and OZ are ****,****,****.**

**Ans. **Let be the direction cosines of a vector equally inclined to axes OX, OY and OZ respectively.

A unit vector along the given vector is

and

……….(i)

Let the given vector (for which unit vector is ) make equal angle (given) (say) with OX OY and OZ

The given vector is in positive octant OXYZ and hence is acute. ……….(ii)

Now angle between and

……….(iii)

Similarly, angle between and , ……….(iv)

And angle between and , ……….(v)

Putting the values of in eq. (i), we get

But [ is acute and hence is positive]

Therefore, required vectors are and

#### NCERT Solutions class 12 Maths Miscellaneous

**12. Let **** and **** Find a vector **** which is perpendicular to both **** and **** and **** **

**Ans. **Given: Vectors and

We know that the cross-product of two vectors, is a vector perpendicular to both and

Hence, vector which is also perpendicular to both and is where or some other scalar.

Therefore,

=

………..(i)

Now given and

Putting in eq. (i), we get

#### NCERT Solutions class 12 Maths Miscellaneous

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

**Ans. **Let , and

Now (say) =

a unit vector along is

=

=

=

…..(i)

Also given Dot product of and is 1.

. = 1

Squaring both sides,

#### NCERT Solutions class 12 Maths Miscellaneous

**14. If **** are mutually perpendicular vectors of equal magnitudes, show that the vector **** is equally inclined to ****.**

**Ans. **Given: are mutually perpendicular vectors of equal magnitude.

……….(i)

And (say) ……….(ii)

Let vector make angles with vectors respectively.

=

= [From eq. (i)]

= = ……….(iii)

We know that

=

Putting the values from eq. (i) and (ii),

= =

Now =

=

Similarly, and

Therefore, is equally inclined to the vectors and

#### NCERT Solutions class 12 Maths Miscellaneous

**15. Prove that **** if and only if **** are perpendicular given **** **

**Ans. **We know that =

= ……….(i)

Now if and are perpendicular

Putting in

= ,

= ……….(ii)

= [Putting value of in eq. (i)]

But (given)

Therefore, vectors and are perpendicular to each other.

#### NCERT Solutions class 12 Maths Miscellaneous

**16. Choose the correct answer:**

**If is the angle between two vectors and then only when:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given:

[ and being lengths of vectors are always 0]

Therefore, option (B) is correct.

#### NCERT Solutions class 12 Maths Miscellaneous

**17. Choose the correct answer:**

**Let and be two unit vectors and is the angle between them. Then is a unit vector if:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given: and are unit vectors.

and

Now squaring both sides of , we have,

, where is the given angle between vectors and .

Putting , we have,

=

= =

Therefore, option (D) is correct.

#### NCERT Solutions class 12 Maths Miscellaneous

**18. Choose the correct answer:**

**The value of is:**

**(A) 0 **

**(B) **

**(C) 1 **

**(D) 3**

**Ans. **

Also = 1 – 1 + 1

Therefore, option (C) is correct.

#### NCERT Solutions class 12 Maths Miscellaneous

**19. If **** be the angle between any two vectors **** and ****, then **** when **** is equal to:**

**(A) 0 **

**(B) **

**(C) **

**(D) **

**Ans. **Given:

And this equation is true only for option (B) namely , since

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Miscellaneous

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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