NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3

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BoardCBSE
TextbookNCERT
ClassClass 12
SubjectMaths
ChapterChapter 10
Chapter NameVector Algebra
ExerciseEx 10.3
Number of Questions Solved18
CategoryNCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3

Ex 10.3 Class 12 Maths Question 1.
Find the angle between two vectors overrightarrow { a } ,overrightarrow { b } with magnitudes √3 and 2 respectively, and such that overrightarrow { a } cdot overrightarrow { b } =sqrt { 6 }
Solution:
Angle θ between two vectors overrightarrow { a } ,overrightarrow { b } ,
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 1

Ex 10.3 Class 12 Maths Question 2.
Find the angle between the vectors hat { i } -2hat { j } +3hat { k } quad andquad 3hat { i } -2hat { j } +hat { k }
Solution:
Let overrightarrow { a } =hat { i } -2hat { j } +3hat { k } quad andquad overrightarrow { b } =3hat { i } -2hat { j } +hat { k }
Let θ be the angle between overrightarrow { a } ,overrightarrow { b } ,
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 2

Ex 10.3 Class 12 Maths Question 3.
Find the projection of the vector overrightarrow { i } -overrightarrow { j } , on the line represented by the vector overrightarrow { i } +overrightarrow { j } ,
Solution:
let overrightarrow { a } =hat { i } -hat { j } quad andquad overrightarrow { b } =hat { i } +hat { j }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 3

Ex 10.3 Class 12 Maths Question 4.
Find the projection of the vector hat { i } +3hat { j } +7hat { k } on the vector 7hat { i } -hat { j } +8hat { k }
Solution:
let overrightarrow { a } =hat { i } +3hat { j } +7hat { k } quad andquad overrightarrow { b } =7hat { i } -hat { j } +8hat { k } then
tiwari academy class 12 maths Chapter 10 Vector Algebra 4

Ex 10.3 Class 12 Maths Question 5.
Show that each of the given three vectors is a unit vector frac { 1 }{ 7 } left( 2hat { i } +3hat { j } +6hat { k } right) ,frac { 1 }{ 7 } left( 3hat { i } -6hat { j } +2hat { k } right) ,frac { 1 }{ 7 } left( 6hat { i } +2hat { j } -3hat { k } right) Also show that they are mutually perpendicular to each other.
Solution:
Letquad overrightarrow { a } =frac { 1 }{ 7 } left( 2hat { i } +3hat { j } +6hat { k } right) ,overrightarrow { b } =frac { 1 }{ 7 } left( 3hat { i } -6hat { j } +2hat { k } right) ,overrightarrow { c } =frac { 1 }{ 7 } left( 6hat { i } +2hat { j } -3hat { k } right)
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 5

Ex 10.3 Class 12 Maths Question 6.
Findleft| overrightarrow { a } right| andleft| overrightarrow { b } right| ifleft( overrightarrow { a } +overrightarrow { b } right) cdot left( overrightarrow { a } -overrightarrow { b } right) =8quad andleft| overrightarrow { a } right| =8left| overrightarrow { b } right|
Solution:
Given left( overrightarrow { a } +overrightarrow { b } right) cdot left( overrightarrow { a } -overrightarrow { b } right) =8
tiwari academy class 12 maths Chapter 10 Vector Algebra 6

Ex 10.3 Class 12 Maths Question 7.
Evaluate the product :
left( 3overrightarrow { a } -5overrightarrow { b } right) cdot left( 2overrightarrow { a } +7overrightarrow { b } right)
Solution:
left( 3overrightarrow { a } -5overrightarrow { b } right) cdot left( 2overrightarrow { a } +7overrightarrow { b } right)
=6overrightarrow { a } .overrightarrow { a } -10overrightarrow { b } overrightarrow { a } +21overrightarrow { a } .overrightarrow { b } -35overrightarrow { b } .overrightarrow { b }
=6{ left| overrightarrow { a } right| }^{ 2 }-11overrightarrow { a } overrightarrow { b } -35{ left| overrightarrow { b } right| }^{ 2 }

Ex 10.3 Class 12 Maths Question 8.
Find the magnitude of two vectors overrightarrow { a } ,overrightarrow { b } having the same magnitude and such that the angle between them is 60° and their scalar product is frac { 1 }{ 2 }
Solution:
We know that overrightarrow { a } .overrightarrow { b } =left| overrightarrow { a } right| left| overrightarrow { b } right| costheta
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 8

Ex 10.3 Class 12 Maths Question 9.
Find left| overrightarrow { x } right| , if for a unit vector overrightarrow { a } ,(overrightarrow { x } -overrightarrow { a } )cdot (overrightarrow { x } +overrightarrow { a } )=12
Solution:
Given
overrightarrow { a } ,(overrightarrow { x } -overrightarrow { a } )cdot (overrightarrow { x } +overrightarrow { a } )=12
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 9

Ex 10.3 Class 12 Maths Question 10.
If overrightarrow { a } =2hat { i } +2hat { j } +3hat { k } ,overrightarrow { b } =-hat { i } +2hat { j } +hat { k } andoverrightarrow { c } =3hat { i } +hat { j } such that overrightarrow { a } +lambda overrightarrow { b } bot overrightarrow { c } , then find the value of λ.
Solution:
Given
overrightarrow { a } =2hat { i } +2hat { j } +3hat { k } ,overrightarrow { b } =-hat { i } +2hat { j } +hat { k } andoverrightarrow { c } =3hat { i } +hat { j }
tiwari academy class 12 maths Chapter 10 Vector Algebra 10

Ex 10.3 Class 12 Maths Question 11.
Show that left| overrightarrow { a } right| overrightarrow { b } +left| overrightarrow { b } right| aquad bot quad left| overrightarrow { a } right| cdot overrightarrow { b } -left| overrightarrow { b } right| a for any two non-zero vectors overrightarrow { a } ,overrightarrow { b }
Solution:
overrightarrow { a } ,overrightarrow { b } are any two non zero vectors
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 11

Ex 10.3 Class 12 Maths Question 12.
If overrightarrow { a } cdot overrightarrow { a } =0quad andquad overrightarrow { a } cdot overrightarrow { b } =0, then what can be concluded about the vector overrightarrow { b } ?
Solution:
overrightarrow { a } overrightarrow { a } =0quad andquad overrightarrow { a } .overrightarrow { b } =0 ,
=> overrightarrow { b } = 0
Hence b is any vector.

Ex 10.3 Class 12 Maths Question 13.
If overrightarrow { a } ,overrightarrow { b } ,overrightarrow { c } are the unit vector such that overrightarrow { a } +overrightarrow { b } +overrightarrow { c } =0 , then find the value of overrightarrow { a } .overrightarrow { b } +overrightarrow { b } .overrightarrow { c } +overrightarrow { c } .overrightarrow { a }
Solution:
We have
overrightarrow { a } +overrightarrow { b } +overrightarrow { c } =0
tiwari academy class 12 maths Chapter 10 Vector Algebra 13

Ex 10.3 Class 12 Maths Question 14.
If either vector overrightarrow { a } =0quad orquad overrightarrow { b } =0 then overrightarrow { a } .overrightarrow { b } =0. But the converse need not be true. Justify your answer with an example.
Solution:
Given: overrightarrow { a } =0quad orquad overrightarrow { b } =0
To prove: overrightarrow { a } .overrightarrow { b } =0
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 14

Ex 10.3 Class 12 Maths Question 15.
If the vertices A,B,C of a triangle ABC are (1,2,3) (-1,0,0), (0,1,2) respectively, then find ∠ABC.
Solution:
Let O be the origin then.
frac { 1 }{ 2 }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 15
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 15.1

Ex 10.3 Class 12 Maths Question 16.
Show that the points A (1,2,7), B (2,6,3) and C (3,10, -1) are collinear.
Solution:
The position vectors of points A, B, C are
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 16

Ex 10.3 Class 12 Maths Question 17.
Show that the vectors 2hat { i } -hat { j } +hat { k } ,hat { i } -3hat { j } -5hat { k } and left( 3hat { i } -4hat { j } -4hat { k } right) from the vertices of a right angled triangle.
Solution:
The position vectors of the points A, B and C are
2hat { i } -hat { j } +hat { k } ,hat { i } -3hat { j } -5hat { k } and left( 3hat { i } -4hat { j } -4hat { k } right)
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 17
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 17.1

Ex 10.3 Class 12 Maths Question 18.
If overrightarrow { a } is a non-zero vector of magnitude ‘a’ and λ is a non- zero scalar, then λ overrightarrow { a } is unit vector if
(a) λ = 1
(b) λ = – 1
(c) a = |λ|
(d) a = frac { 1 }{ left| lambda right| }
Solution:
left| overrightarrow { a } right| =a
Given : lambda overrightarrow { a } is a unit vectors
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 18

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