Class 12 Maths NCERT Solutions for Chapter 10 Vector Algebra Exercise 10.3
Vector Algebra Exercise 10.3 Solutions
1. Find the angle between two vectors a⃗ and b⃗ with magnitudes √3 and 2, respectively having a⃗ . b⃗ = √6.
Solution
It is given that,
| a⃗ | = √3, | b⃗ | = 2 and, a⃗ . b⃗ = √6 .
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhn9EW8NuZ35orEi28T1ItLg6tqtaK7Q0qki4ZsSjlsHYIw6ludCaBU7MvMyy3kSvs7Nx9Ao53n82UohHh3o_ULKKwwEhAfS3vpw2CIV5-W7-qZAeLo0WOT65qklbuqpL1DPR8as9gqusyTwyJmiEFAm5BbVFhD-oRg-uqJ_MdAtSdFY5_wl7rv42P/w261-h200-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%201.JPG)
Hence, the angle between the given vectors a⃗ and b⃗ is Ï€/4.
2. Find the angle between the vectors i ^ - 2j ^ + 3k^ and 3i ^ - 2j ^ + k^ .
Solution
The given vectors are a⃗ = i ^ - 2j ^ + 3k^ and b⃗ = 3i ^ - 2j ^ + k^ . ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje-pBViPqJsm5pYJ_pJmsK8yKS_oNrfYzOKkbRsLQSJuLopqHBO1NXnXjSC2XzZz0bLoj17zX5YpyGeIB9FXoryF1HfWS3HESDDYXX7srcnNxKSNvf-K-Y1GiDQQntT9ZzhXgi4UmzknnwH2qkM-kHQxFtq2bVpTuYgi5uGvzJatJHqBMqj8HrPjm6/w409-h379-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%202.JPG)
3. Find the projection of the vector i ^ - j ^ on the vector i ^ + j ^.
Solution
4. Find the projection of the vector i ^ + 3j ^ + 7k^ on the vector 7i ^ - j ^ + 8k^ .
Solution
5. Show that each of the given three vectors is a unit vector :
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjqow3B9ua7ONBxTynDf4IdJHKLSzl2Uf4onuRVFN2G4FemFXE0zqmLc5-tdICdiTCtcwHPXbBufUsqXVBCgV7_M2F1RG80r_ZS3UD4tJC1qh0c3yG4Yg8-ONR3cuIm_kta1vRXy8ocYAzEL5MJgXp95TSsLxVvihHXTgzpOrBwnwGQ4QDJde1eXLS/w341-h41-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%205.JPG)
Also, show that they are mutually perpendicular to each other.Solution
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMam_25zS9emeqNh4Dp-OtKRGO8EnwWbMwXiCKv8JlxUqBiQ5yQtLxWPALhHmv9VIubT2-eY4GrA8jen51vZ08M60BUfAj6rfnTHEApCqXulJBeyVgSDCZD_GXacbgIY-RTQXwwAEATTMOkwtLmHTGqIhK2IBLSuWEso7s9JSaK7FkiE-ioJ_K8vaf/w357-h332-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%206.JPG)
Thus, each of the given three vectors is a unit vector.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcwJ8XxGQV0ucyg29Rwj4yBE2r-xPuOoNlG_afViZuky23kdSkzy8n1IFPkajJtmnqvQqZbbrQXiOE-1Nr3ZKGpkTjCKYmoel9vTdFR0s09DkTKgp7M1L7Wwy7J-LMqHp_SrWUeAygg15EH72qwjDLCxX-Aee62FToMUTcLPZwSFnH5Sxcshy74Rea/w380-h153-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%207.JPG)
Hence, the given three vectors are mutually perpendicular to each other.
6. Find | a⃗ | and | b⃗ |, if ( a⃗ + b⃗) . ( a⃗ - b⃗ ) = 8 and |a⃗ | = 8| b⃗ |.
Solution
7. Evaluate the product ( 3a⃗ - 5b⃗ ) . ( 2a⃗ + 7b⃗ ).
Solution
8. Find the magnitude of two vectors a⃗ and b⃗ , having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2 .
Solution
Let θ be the angle between the vectors a⃗ and b⃗ .
It is given that | a⃗ | = | b⃗ |, a⃗ . b⃗ = 1/2 , and θ = 60° ...(1)
We know that a⃗ . b⃗ = | a⃗ | | b⃗ | cos θ.
9. Find | x⃗ |, if for a unit vector a⃗ , ( x⃗ - a⃗ ).( x⃗ + a⃗ ) = 12.
Solution
10. If a⃗ = 2i ^ - 2j ^ + 3k^ , b⃗ = - i ^ + 2j ^ + k^ and c⃗ = 3i ^ + j ^ are such that a⃗ = λb⃗ is perpendicular c⃗ , then find the value of λ .
Solution
Hence, the required value of λ is 8.
11. Show that (|a⃗ |b⃗ ) + (|b⃗ |a⃗ )is perpendicular to (|a⃗ |b⃗ ) - (|b⃗ |a⃗), for any two nonzero vectors a⃗ and b⃗ .
Solution
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipExwQ0ZfcMMB8NdX_Zg6tbj2kQuMg0HCXYQ_nDPjUgIiwHQQrGyBi1IRhVSmtBDQ92L29PR3C4ZhFBAPy0MZFNq0O4uebeZ2oFpnvNZ391Rbtuo3BJNAzMEb0hps2zHHhIzmjDM3mLkEOcwsmMzlKlICjOJvApddN05XP410wezsQgoH1VfCcJT0s/w314-h145-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%2013.JPG)
Hence, | a
⃗ |b
⃗ + | b
⃗ |a
⃗ and | a
⃗ |b
⃗ - | b
⃗ |a
⃗ are perpendicular to each other.
12. If a⃗ . a⃗ = 0 and a⃗ . b⃗ = 0, then what can be concluded about the vector b⃗ ?
Solution
It is given that a⃗. a⃗ = 0 and a⃗ . b⃗ = 0
Now,
Hence, vector b⃗ satisfying a⃗ . b⃗ = 0 can be any vector.
13. If a⃗ , b⃗ and c⃗ are unit vectors such that a⃗ + b⃗ + c⃗ = 0⃗ , find the value of a⃗ . b⃗ + b⃗ . c⃗ + c⃗ . a⃗ .
Solution
14. If either vector a⃗ = 0⃗ or b⃗ = 0⃗ then a⃗ , b⃗ = 0. But the converse need not be true. justify your answer with an example.
Solution
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBY9ifLQBZwh57rqzaZ4_ooq8pegj51OODx_hobXjslLAwnVQT3_Z2t5JXHyxmDpr5OCWGWf_Y20NJKnhj-h9AKc6-mLsq-ateXl3X5VzE_zx_rSmW51LQJL6ik6Qm9CqkD0wYH0Cn25nnc0nG_eEjpL9ralP2ySVjRpz49fKH9NtZgWCVUXUegZIP/w322-h270-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%2016.JPG)
Hence, the converse of the given statement need not be true.
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. [∠ABC is the angle between the vectors and BA⃗ and BC⃗ ]
Solution
The vertices of ΔABC are given as A (1, 2, 3), B (–1, 0, 0), and C (0, 1, 2).
Also, it is given that ∠ABC is the angle between the vectors BA⃗ and BC⃗ .
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYpTlHH5lo4JHyWOH_Dm7-1spB4m0bd4jMBf4AtItgs2wN5d9IZt1Jjvu24tL8mvqg6PaxjDvQ0kx82TXrnJ5znOt7praWkj5kPl7u9FBWPvtjyltENVAnxEzdds4vcM4rFcdBU56C5rW-wWsDY7_dh5Nd0uw5mqVOpxglIxuWlIPDXKTY7aEQH3VC/w478-h409-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%2017.JPG)
16. Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.
Solution
The given point are A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) .
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUCiKcZUDvhOj4EIeyXqeGU7p1TInOtvTSoZA6ZKHhg1sY3kMHxajvisbW9gKSrFEJLISY-L8LvJEX1o5L3OkE8ko_GlfJPP2v3tbvWnQ3p9KjJzCKAUUCyB5di2QlcIOqgJwmlNGUGU15oNc_2fAFE1zGH7RAkxVQw04F_YvHhdAYqg-jsnb4hnRG/w356-h250-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%2018.JPG)
Hence, the given points A, B, and C are collinear.
17. Show that the vectors 2i ^- j ^+k^ , i ^-3j ^-5k^ and 3i ^- 4j ^-4k^ from the vertices of a right angled triangle.
Solution
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6qDXcgmzXLKevH8y3Ytou0Cxo0RoUaO7eo-GhyvQ2L8LiLla0-yrKnT1EVaC4g4TDKjA6CDSkVTZ-ggozWoePBPr-UjHW8eooyOumA9fQ0iGh-v2O86iAUrZSvv0C6OJW84OtWibOJ_4TLpODSbmegmAWNf4fd0oa72L6arufLLZGruG3UZHLuX1T/w602-h439-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%2019.JPG)
Hence, Δ ABC is a right angled triangle.
18. If a⃗ is a nonzero vector of magnitude 'a' and λa⃗ is unit vector if
(A) λ = 1
(B) λ = -1
(C) a = |λ|
(D) a = 1/|λ|
Solution
Vector λa⃗ is a unit vector if |λa⃗| = 1 .
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhTgPyv4rNXT5yxBslEIn1WFZkeUAzyY7FbEjpZZWDHVtAZmFeTBdwPHL6pHDA2WJYDKwfJgOWQZdu1emFJ5UkAgJAmQKpJ_ejILB4uj46GmOYMatNTL43v7WfXfF5QyJ4vf4PygJyFWHoFPICKS35dJr3Hb6wXtLGzPFFmqr9Xh0QpijF-1YLB3m/s1600-rw/NCERT%20Solutions%20for%20Chapter%2010%20Vector%20Algebra%20Class%2012%20Maths%20Exercise%2010.3%20img%2020.JPG)
Hence, vector λa
⃗ is a unit vector if a = 1/|λ|.
The correct answer is D.