NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2

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BoardCBSE
TextbookNCERT
ClassClass 12
SubjectMaths
ChapterChapter 10
Chapter NameVector Algebra
ExerciseEx 10.2
Number of Questions Solved19
CategoryNCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2

Ex 10.2 Class 12 Maths Question 1.
Compute the magnitude of the following vectors:
overrightarrow { a } =hat { i } +hat { j } +hat { k } ,overrightarrow { b } =hat { 2i } -hat { 7j } -hat { 3k }
overrightarrow { c } =frac { 1 }{ sqrt { 3 } } hat { i } +frac { 1 }{ sqrt { 3 } } hat { j } -frac { 1 }{ sqrt { 3 } } hat { k }
Solution:
overrightarrow { a } =hat { i } +hat { j } +hat { k }
left| overrightarrow { a } right| =sqrt { { 1 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 1
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 1.1

Ex 10.2 Class 12 Maths Question 2.
Write two different vectors having same magnitude.
Solution:
overrightarrow { a } =hat { i } +hat { 2j } +hat { 3k } ,overrightarrow { b } =hat { 3i } +hat { 2j } +hat { k }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 2
Such possible answers are infinite

Ex 10.2 Class 12 Maths Question 3.
Write two different vectors having same direction.
Solution:
Let the two vectors be
overrightarrow { a } =hat { i } +hat { j } +hat { k } ,overrightarrow { b } =hat { 3i } +hat { 3j } +hat { 3k }
tiwari academy class 12 maths Chapter 10 Vector Algebra 3
Hence vectors overrightarrow { a } ,overrightarrow { b } have the same direction but different magnitude

Ex 10.2 Class 12 Maths Question 4.
Find the values of x and y so that the vectors overrightarrow { 2i } +overrightarrow { 3j } quad andquad hat { xi } +hat { yj } are equal.
Solution:
We are given overrightarrow { 2i } +overrightarrow { 3j } quad andquad hat { xi } +hat { yj }
If vectors are equal, then their respective components are equal. Hence x = 2, y = 3.

Ex 10.2 Class 12 Maths Question 5.
Find the scalar and vector components of the vector with initial point (2,1) and terminal point (-5,7).
Solution:
LetA(2, 1) be the initial point and B(-5,7) be the terminal point overrightarrow { AB } =left( { x }_{ 2 }-{ x }_{ 1 } right) hat { i } +left( { y }_{ 2 }-{ y }_{ 1 } right) hat { j } =-hat { 7i } +hat { 6j }
∴The vector components are -hat { 7i } andhat { 6j } and scalar components are – 7 and 6.

Ex 10.2 Class 12 Maths Question 6.
Find the sum of three vectors:
overrightarrow { a } =hat { i } -hat { 2j } +hat { k } ,overrightarrow { b } =-2hat { i } +hat { 4j } +5hat { k } quad andquad overrightarrow { c } =hat { i } -hat { 6j } -hat { 7k } ,
Solution:
overrightarrow { a } =hat { i } -hat { 2j } +hat { k } ,overrightarrow { b } =-2hat { i } +hat { 4j } +5hat { k } quad andquad overrightarrow { c } =hat { i } -hat { 6j } -hat { 7k } ,
overrightarrow { a } +overrightarrow { b } +overrightarrow { c } =hat { 0i } -hat { 4j } -hat { k } =-4hat { i } -hat { k }

Ex 10.2 Class 12 Maths Question 7.
Find the unit vector in the direction of the vector
overrightarrow { a } =hat { i } +hat { j } +hat { 2k }
Solution:
overrightarrow { a } =hat { i } +hat { j } +hat { 2k }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 7

Ex 10.2 Class 12 Maths Question 8.
Find the unit vector in the direction of vector overrightarrow { PQ } , where P and Q are the points (1,2,3) and (4,5,6) respectively.
Solution:
The points P and Q are (1, 2, 3) and (4, 5, 6) respectively
overrightarrow { PQ } =(4-1)hat { i } +(5-2)hat { j } +(6-3)hat { k }
tiwari academy class 12 maths Chapter 10 Vector Algebra 8

Ex 10.2 Class 12 Maths Question 9.
For given vectors overrightarrow { a } =2hat { i } -hat { j } +2hat { k } quad andquad overrightarrow { b } =-hat { i } +hat { j } -hat { k } find the unit vector in the direction of the vector overrightarrow { a } +overrightarrow { b }
Solution:
overrightarrow { a } =2hat { i } -hat { j } +2hat { k } quad andquad overrightarrow { b } =-hat { i } +hat { j } -hat { k }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 9

Ex 10.2 Class 12 Maths Question 10.
Find a vector in the direction of 5hat { i } -hat { j } +2hat { k } which has magnitude 8 units.
Solution:
The given vector is overrightarrow { a } =5hat { i } -hat { j } +2hat { k }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 10

Ex 10.2 Class 12 Maths Question 11.
Show that the vector 2hat { i } -3hat { j } +4hat { k } quad andquad -4hat { i } +6hat { j } -8hat { k } are collinear.
Solution:
overrightarrow { a } =2hat { i } -3hat { j } +4hat { k } quad andquad overrightarrow { b } =-4hat { i } +6hat { j } -8hat { k }
=-2(2hat { i } -3hat { j } +4hat { k } )
vector overrightarrow { a } quad andquad overrightarrow { b } have the same direction they are collinear.

Ex 10.2 Class 12 Maths Question 12.
Find the direction cosines of the vector hat { i } +2hat { j } +3hat { k }
Solution:
let overrightarrow { p } =hat { i } +2hat { j } +3hat { k }
Now a = 1,b = 2,c = 3
tiwari academy class 12 maths Chapter 10 Vector Algebra 12

Ex 10.2 Class 12 Maths Question 13.
Find the direction cosines of the vector joining the points A (1,2, -3) and B(-1, -2,1), directed fromAtoB.
Solution:
Vector joining the points A and B is
({ x }_{ 2 }-{ x }_{ 1 })hat { i } +({ y }_{ 2 }-{ y }_{ 1 })hat { j } +({ z }_{ 2 }-{ z }_{ 1 })hat { k }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 13

Ex 10.2 Class 12 Maths Question 14.
Show that the vector hat { i } +hat { j } +hat { k } are equally inclined to the axes OX, OY, OZ.
Solution:
Let hat { i } +hat { j } +hat { k } =overrightarrow { a } , Direction cosines of vector xhat { i } +yhat { j } +zhat { k } are
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 14
which shows that the vector a is equally inclined to the axes OX, OY, OZ.

Ex 10.2 Class 12 Maths Question 15.
Find the position vector of a point R which divides the line joining the points whose positive vector are P(hat { i } +2hat { j } -hat { k } )quad andquad Q(-hat { i } +hat { j } +hat { k } ) in the ratio 2:1
(i) internally
(ii) externally.
Solution:
(i) The point R which divides the line joining the point P(overrightarrow { a } )quad andquad Q(overrightarrow { b } ) in the ratio m : n
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 15
tiwari academy class 12 maths Chapter 10 Vector Algebra 15.1

Ex 10.2 Class 12 Maths Question 16.
Find position vector of the mid point of the vector joining the points P (2,3,4) and Q (4,1, -2).
Solution:
Let overrightarrow { OP } =2hat { i } +3hat { j } +4hat { k } quad andquad overrightarrow { OQ } =4hat { i } +hat { j } -2hat { k }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 16

Ex 10.2 Class 12 Maths Question 17.
Show that the points A, B and C with position vector overrightarrow { a } =3hat { i } -4hat { j } -4hat { k } ,overrightarrow { b } =2hat { i } -hat { j } +hat { k } andquad overrightarrow { c } =hat { i } -3hat { j } -5hat { k } respectively form the vertices of a right angled triangle.
Solution:
overrightarrow { AB } =overrightarrow { b } -overrightarrow { a } =-hat { i } +3hat { j } +5hat { k }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 17
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 17.1

Ex 10.2 Class 12 Maths Question 18.
In triangle ABC (fig.), which of the following is not
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 18
(a) overrightarrow { AB } +overrightarrow { BC } +overrightarrow { CA } =overrightarrow { 0 }
(b) overrightarrow { AB } +overrightarrow { BC } -overrightarrow { AC } =overrightarrow { 0 }
(c) overrightarrow { AB } +overrightarrow { BC } -overrightarrow { CA } =overrightarrow { 0 }
(d) overrightarrow { AB } -overrightarrow { CB } +overrightarrow { CA } =overrightarrow { 0 }
Solution:
We know that
overrightarrow { AB } +overrightarrow { BC } +overrightarrow { CA } =overrightarrow { 0 }
overrightarrow { AB } +overrightarrow { BC } -overrightarrow { AC } =overrightarrow { 0 }
Hence option (c) is not correct

Ex 10.2 Class 12 Maths Question 19.
If overrightarrow { a } ,overrightarrow { b } are two collinear vectors then which of the following are incorrect:
(a) overrightarrow { b } =lambda overrightarrow { a } , for some scalar λ.
(b) overrightarrow { a } =pm overrightarrow { b }
(c) the respective components of overrightarrow { a } ,overrightarrow { b } are proportional.
(d) both the vectors overrightarrow { a } ,overrightarrow { b } have same direction, but different magnitudes.
Solution:
Options (d) is incorrect since both the vectors overrightarrow { a } ,overrightarrow { b } , being collinear, are not necessarily in the same direction. They may have opposite directions. Their magnitudes may be different.

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