NCERT Solutions for Miscellaneous Exercise Chapter 10 Class 12 - Vector Algebra

NCERT Solutions for Miscellaneous Exercise Chapter 10 Class 12 - Vector Algebra

Komal MiglaniUpdated on 07 May 2025, 05:08 PM IST

Curious to know how well you have learned Vector Algebra? The NCERT Solutions for Class 12 maths chapter 10 miscellaneous exercise are ideal to test your knowledge with mixed and challenging questions. These 19 questions cover all that you have learned till now in this chapter, and help you revise and increase your confidence level before exams. These solutions are prepared by Careers360 experts, these solutions are according to the latest CBSE syllabus.

Attempt to solve the Class 12 Maths Chapter 10 Miscellaneous Exercise of NCERT yourself to gauge how well you have understood the concepts. These questions are a combination of all important topics and assist you in analyzing what needs to be revised. The NCERT solution for Class 12 Maths are aimed at building your foundation through step-by-step solutions.

This Story also Contains

  1. NCERT Solutions Class 12 Maths Chapter 10: Miscellaneous Exercise
  2. NCERT Solutions Subject Wise
  3. Subject Wise NCERT Exemplar Solutions

NCERT Solutions Class 12 Maths Chapter 10: Miscellaneous Exercise

Question 1: Write down a unit vector in XY-plane, making an angle of 30 with the positive direction of x-axis.

Answer:

As we know

a unit vector in XY-Plane making an angle θ with x-axis :

r=cosθi^+sinθj^

Hence for θ=300

r=cos(300)i^+sin(300)j^

r=32i^+12j^

Answer- the unit vector in XY-plane, making an angle of 30 with the positive direction of x-axis is

r=32i^+12j^

Question 2: Find the scalar components and magnitude of the vector joining the points
P(x1,y1,z1)andQ(x2,y2,z2).

Answer:

Given in the question

P(x1,y1,z1)andQ(x2,y2,z2).

And we need to finrd the scalar components and magnitude of the vector joining the points P and Q

PQ=(x2x1)i^+(y2y1)j^+(z2z1)k^

Magnitiude of vector PQ

|PQ|=(x2x1)2+(y2y1)2+(z2z1)2

Scalar components are

(x2x1),(y2y1),(z2z1)

Question 3: A girl walks 4 km towards west, then she walks 3 km in a direction 30 east of north and stops. Determine the girl’s displacement from her initial point of departure.

Answer:

As the girl walks 4km towards west

Position vector = 4i^

Now as she moves 3km in direction 30 degree east of north.

4i^+3sin300i^+3cos300j^

4i^+32i^+332j^

52i^+332j^

hence final position vector is;

52i^+332j^

Question 4: If a=b+c , then is it true that |a|=|b|+|c| ? Justify your answer.

Answer:

No, if a=b+c then we can not conclude that |a|=|b|+|c| .

the condition a=b+c satisfies in the triangle.

also, in a triangle, |a|<|b|+|c|

Since, the condition |a|=|b|+|c| is contradicting with the triangle inequality, if a=b+c then we can not conclude that |a|=|b|+|c|

Question 5: Find the value of x for which x(i^+j^+k^) is a unit vector.

Answer:

Given in the question,

a unit vector, u=x(i^+j^+k^)

We need to find the value of x

|u|=1

|x(i^+j^+k^)|=1

x12+12+12=1

x3=1

x=13

The value of x is 13

Question 6: Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a=2i^+3j^k^andb=i^2j^+k^

Answer:

Given two vectors

a=2i^+3j^k^andb=i^2j^+k^

Resultant of a and b :

R=a+b =2i^+3j^k^+i^2j^+k^=3i^+j^

Now, a unit vector in the direction of R

u=3i^+j^32+12=310i^+110j^

Now, a unit vector of magnitude in direction of R

v=5u=5310i^+5110j^=1510i^+510j^

Hence the required vector is 1510i^+510j^

Question 7: If a=i^+j^+k^,b=2i^j^+3k^andc=i^2j^+k^ , find a unit vector parallel to the vector 2ab+3c .

Answer:

Given in the question,

a=i^+j^+k^,b=2i^j^+3k^andc=i^2j^+k^

Now,

let vector V=2ab+3c

V=2(i^+j^+k^)(2i^j^+3k^)+3(i^2j^+k^)

V=3i^3j^+2k^

Now, a unit vector in direction of V

u=3i^3j^+2k^32+(3)2+22=322i^322j^+222k^

Now,

A unit vector parallel to V

u=322i^322j^+222k^

OR

u=322i^+322j^222k^

Question 8: Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

Answer:

Given in the question,

points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)

AB=(51)i^+(0(2))j^+(2(8))k^=4i^+2j^+6k^

BC=(115)i^+(30)j^+(7(2))k^=6i^+3j^+9k^

CA=(111)i^+(3(2))j^+(7(8))k^=10i^+5j^+15k^

now let's calculate the magnitude of the vectors

|AB|=42+22+62=56=214

|BC|=62+32+92=126=314

|CA|=102+52+152=350=514

As we see that AB = BC + AC, we conclude that three points are colinear.

we can also see from here,

Point B divides AC in the ratio 2 : 3.

Question 9: Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a+b)and(a3b) externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Answer:

Given, two vectors P=(2a+b)andQ=(a3b)

the point R which divides line segment PQ in ratio 1:2 is given by

=2(2a+b)(a3b)21=4a+2ba+3b=3a+5b

Hence position vector of R is 3a+5b .

Now, Position vector of the midpoint of RQ

=(3a+5b+a3b)2=2a+b

which is the position vector of Point P . Hence, P is the mid-point of RQ

Question 10: The two adjacent sides of a parallelogram are 2i^4j^+5k^andi^2j^3k^ . Find the unit vector parallel to its diagonal. Also, find its area.

Answer:

Given, two adjacent sides of the parallelogram

2i^4j^+5k^andi^2j^3k^

The diagonal will be the resultant of these two vectors. so

resultant R:

R=2i^4j^+5k^+i^2j^3k^=3i^6j^+2k^

Now unit vector in direction of R

u=3i^6j^+2k^32+(6)2+22=3i^6j^+2k^49=3i^6j^+2k^7

Hence unit vector along the diagonal of the parallelogram

u=37i^67j^+27k^

Now,

Area of parallelogram

A=(2i^4j^+5k^)×(i^2j^3k^)

A=|i^j^k^245123|=|i^(12+10)j^(65)+k^(4+4)|=|22i^+11j^|

A=222+112=115

Hence the area of the parallelogram is 115 .

Question 11: Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ±(13,13,13)

Answer:

Let a vector a is equally inclined to axis OX, OY and OZ.

let direction cosines of this vector be

cosα,cosαandcosα

Now

cos2α+cos2α+cos2α=1

cos2α=13

cosα=13

Hence direction cosines are:

(13,13,13)

Question 12: Let a=i^+4j^+2k^,b=3i^2j^+7k^andc=2i^j^+4k^ . Find a vector d which is perpendicular to both aandbandc.d=15

Answer:

Given,

a=i^+4j^+2k^,b=3i^2j^+7k^andc=2i^j^+4k^

Let d=d1i^+d2j^+d3k^

now, since it is given that d is perpendicular to a and b , we got the condition,

b.d=0 and a.d=0

(i^+4j^+2k^)(d1i^+d2j^+d3k^)=0
And
(3i^2j^+7k^)(d1i^+d2j^+d3k^)=0

d1+4d2+2d3=0 And 3d12d2+7d3=0

here we got 2 equation and 3 variable. one more equation will come from the condition:

c.d=15

(2i^j^+4k^)(d1i^+d2j^+d3k^)=15

2d1d2+4d3=15

so now we have three equations and three variables,

d1+4d2+2d3=0

3d12d2+7d3=0

2d1d2+4d3=15

On solving this three equation we get,

d1=1603,d2=53andd3=703 ,

Hence required vector :

d=1603i^53j^703k^

Question 13: The scalar product of the vector i^+j^+k^ with a unit vector along the sum of vectors 2i^+4j^5k^ and λi^+2j^+3k^ is equal to one. Find the value of λ.

Answer:

Let, the sum of vectors 2i^+4j^5k^ and λi^+2j^+3k^ be

a=(λ+2)i^+6j^2k^

unit vector along a

u=(λ+2)i^+6j^2k^(λ+2)2+62+(2)2=(λ+2)i^+6j^2k^λ2+4λ+44

Now, the scalar product of this with i^+j^+k^

u.(i^+j^+k^)=(λ+2)i^+6j^2k^λ2+4λ+44.(i^+j^+k^)

u.(i^+j^+k^)=(λ+2)+62λ2+4λ+44=1

(λ+2)+62λ2+4λ+44=1

(λ+6)λ2+4λ+44=1

λ=1

Question 14: If a,b,c are mutually perpendicular vectors of equal magnitudes, show that the vector a+b+c is equally inclined to a,bandc .

Answer:

Given

|a|=|b|=|c| and

a.b=b.c=c.a=0

Now, let vector a+b+c is inclined to a,bandc at θ1,θ2andθ3 respectively.

Now,

cosθ1=(a+b+c).a|a+b+c||a|=a.a+a.b+c.a|a+b+c||a|=a.a|a+b+c||a|=|a||a+b+c|

cosθ2=(a+b+c).b|a+b+c||b|=a.b+b.b+c.b|a+b+c||b|=b.b|a+b+c||b|=|b||a+b+c|

cosθ3=(a+b+c).c|a+b+c||c|=a.c+b.c+c.c|a+b+c||c|=c.c|a+b+c||c|=|c||a+b+c|

Now, Since, |a|=|b|=|c|

cosθ1=cosθ2=cosθ3

θ1=θ2=θ3

Hence vector a+b+c is equally inclined to a,bandc .

Question 15: Prove that (a+b).(a+b)=|a2|+|b|2 , if and only if a,b are perpendicular, given a0,b0

Answer:

Given in the question,

a,b are perpendicular and we need to prove that (a+b)(a+b)=|a2|+|b|2 LHS =(a+b)(a+b)=aa+ab+ba+bb=aa+2ab+bb=|a|2+2ab+|b|2 if a,b are perpendicular, ab=0(a+b)(a+b)=|a|2+2ab+|b|2=|a|2+|b|2

= RHS

LHS ie equal to RHS

Hence proved.

Question 16: Choose the correct answer If θ is the angle between two vectors aandb , then ab0 only when
A)0<θ<π2B)0θπ2C)0<θ<πD)0θπ

Answer:

Given in the question

θ is the angle between two vectors aandb

ab0

|a||b|cosθ0

this will satisfy when

cosθ0

0θπ2

Hence option B is the correct answer.

A)θ=π4B)θ=π3C)θ=π2D)θ=2π3

Answer:

Gicen in the question

aandb be two unit vectors and θ is the angle between them

|a|=1,and|b|=1

also

|a+b|=1

|a+b|2=1

|a|2+|b|2+2a.b=1

1+1+2a.b=1

a.b=12

|a||b|cosθ=12

cosθ=12

θ=2π3

Then a+b is a unit vector if θ=2π3

Hence option D is correct.

Question 18: The value of i^(j^×k^)+j^(i^×k^)+k^(i^×j^) is

(A) 0

(B) –1

(C) 1

(D) 3

Answer:

To find the value of i^(j^×k^)+j^(i^×k^)+k^(i^×j^)

i^(j^×k^)+j^(i^×k^)+k^(i^×j^)=i^.i^+j^(j^)+k^.k^=11+1=1

Hence option C is correct.

Question 19: Choose the correct. If θ is the angle between any two vectors aandb , then |ab|=|a×b| when θ
is equal to

(A) 0
(B) π/4
( C )π/2
( D ) π

Answer:

Given in the question

θ is the angle between any two vectors aandb and |ab|=|a×b|

To find the value of θ

Hence, option D is correct.

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Frequently Asked Questions (FAQs)

Q: Why should I study Class 12 Maths chapter Vectors?
A:

Vectors is an important chapter of Class 12 Maths NCERT syllabus. The concepts studied in this chapter are used not only on mathematics but also to solve problems in Class 11 and 12 physics problems also. Also Vectors is an important chapter for CBSE Board exam. Solving the NCERT questions and referring to the NCERT exemplar will be beneficial for board exam preparation.

Q: What are the topics coverd under Vectors Class 12 Maths NCERT syllabus?
A:

The topics covered are vectors basic concepts, algebra of vectors, direction cosine and ratios, dot product and cross product of vectors.

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Yes, in the case that you appeared for the 2025 improvement exam and your roll number is different from what was on the previous year’s marksheet, the board will usually release a new migration certificate. This is because the migration certificate will reflect the most recent exam details, roll number and passing year. You can apply to get it from your board using the process prescribed by them either online or through your school/college.

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From the academic year 2026, the board will conduct the CBSE 10th exam twice a year, while the CBSE 12th exam will be held once, as per usual. For class 10th, the second phase exam will act as the supplementary exam. Check out information on w hen the CBSE first exam 2026 will be conducted and changes in 2026 CBSE Board exam by clicking on the link .

If you want to change your stream to humanities after getting a compartment in one subject in the CBSE 12th Board Exam , you actually have limited options to qualify for your board exams. You can prepare effectively and appear in the compartment examination for mathematics again. If you do not wish to continue with the current stream, you can take readmission in the Humanities stream and start from Class 11th again, and continue studying for two more years to qualify for the 12th examination.

The GUJCET Merit List is prepared based on the Class 12th marks and GUJCET marks received by the students. CBSE students who are not from the Gujarat board can definitely compete with GSEB students, as their eligibility is decided based on the combined marks scored by them in GUJCET and the 12th board. The weightage of the GUJCET score is 40% and the weightage of the class 12 scores is 60%.