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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.2

11th Maths Exercise 8.2 Question 1.
Verify whether the following ratios are direction cosines of some vector or not.
11th Maths Exercise 8.2 Solutions Chapter 8 Vector Algebra - I Samacheer Kalvi
Solution:
11th Maths Vector Algebra Solutions Chapter 8 Ex 8.2 Samacheer Kalvi

11th Maths Vector Algebra Solutions Question 2.
Find the direction cosines of a vectors whose direction ratios are
(i) 1, 2, 3
(ii) 3, -1, 3
(iii) 0, 0, 7
Solution:
11th Maths Exercise 8.2 Answers Solutions Chapter 8 Vector Algebra - I Samacheer Kalvi

11th Maths Exercise 8.2 Answers Question 3.
Find the direction cosines and direction ratios for the following vectors
11th Maths Vector Algebra Exercise 8.2 Chapter 8 Samacheer Kalvi
Solution:
11th Maths 8.2 Solutions Chapter 8 Vector Algebra Samacheer Kalvi
Class 11th Maths Exercise 8.2 Solution Chapter 8 Vector Algebra Samacheer Kalvi
Class 11 Maths Chapter 8 Exercise 8.2 Solutions Vector Algebra Samacheer Kalvi

11th Maths Vector Algebra Exercise 8.2 Question 4.
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians.
Solution:
11th Maths Guide Solutions Chapter 8 Vector Algebra - I Ex 8.2 Samacheer Kalvi
Samacheer Guru 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2
11th Samacheer Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 Samacheer Kalvi

11th Maths 8.2 Question 5.
If \(\frac{1}{2}, \frac{1}{\sqrt{2}}\), a are the direction cosines of some vector, then find a.
Solution:
Samacheer Kalvi 11th Guide Maths Chapter 8 Vector Algebra - I Ex 8.2

Class 11th Maths Exercise 8.2 Solution Question 6.
If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c.
Solution:
Let A be the point (1, 0, 0) and B be the point (0, 1, 0) (i.e.,) \(\overrightarrow{\mathrm{OA}}=\hat{i}\) and \(\overrightarrow{\mathrm{OB}}=\hat{j}\)
Then \(\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\hat{j}-\hat{i}=-\hat{i}+\hat{j}\)
= (-1, 1, 0)
= (a, a + b, a + b + c)
⇒ a = -1, a + b = 1 and a + b + c = 0
Now a = -1 ⇒ -1 + b = 1 ;a + b + c = 0
⇒ b = 2; -1 + 2 + c = 0 ⇒ c + 1 = 0
⇒ c = -1
∴ a = -1; b = 2; c = -1.
Note: If we taken \(\overrightarrow{\mathrm{BA}}\) then we get a = 1, b = -2 and c = 1.

Class 11 Maths Chapter 8 Exercise 8.2 Solutions Question 7.
Show that the vectors \(2 \hat{i}-\hat{j}+\hat{k}, 3 \hat{i}-4 \hat{j}-4 \hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}\) form a right angled triangle.
Sol:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2
⇒ The given vectors form the sides of a right angled triangle.

11th Maths Guide Question 8.
Find the value of k for which the vectors \(\vec{a}=3 \hat{i}+2 \hat{j}+9 \hat{k}\) and \(\vec{b}=\hat{i}+\lambda \hat{j}+3 \hat{k}\) are parallel.
Solution:
11 Samacheer Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2

Samacheer Guru 11th Maths Question 9.
Show that the following vectors are coplanar.
Ex 8.2 Class 11 Maths Solutions Chapter 8 Vector Algebra Samacheer Kalvi
Solution:
Let the given three vectors be \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\). When we are able to write one vector as a linear combination of the other two vectors, then the given vectors are called coplanar vectors.
11th Maths 8th Chapter Solutions Vector Algebra - I Ex 8.2 Samacheer Kalvi
Samacheer Kalvi Maths Guide 11th Solutions Chapter 8 Vector Algebra - I Ex 8.2
Samacheer Kalvi.Guru 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2
Maths Guide 11th Solutions Chapter 8 Vector Algebra - I Ex 8.2 Samacheer Kalvi
We are able to write \(\vec{a}\) as a linear combination of \(\vec{b}\) and \(\vec{c}\)
∴ The vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are coplanar

11th Samacheer Maths Solutions Question 10.
Show that the points whose position vectors Maths Solutions Class 11 Samacheer Kalvi Chapter 8 Vector Algebra - I Ex 8.2 and Samacheer Kalvi 11th Maths Solution Book Chapter 8 Vector Algebra - I Ex 8.2 are coplanar
Solution:
Let the given points be A, B, C and D. To prove that the points A, B, C, D are coplanar, we have to prove that the vectors \(\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}\) and \(\overrightarrow{\mathrm{AC}}\) are coplanar
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 20
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 21
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 22
∴ we are able to write one vector as a linear combination of the other two vectors ⇒ the given vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are coplanar.
(i.e.,) the given points A, B, C, D are coplanar.

Samacheer Kalvi 11th Guide Maths Question 11.
If \(\vec{a}=2 \hat{i}+3 \hat{j}-4 \hat{k}\), \(\vec{b}=3 \hat{i}-4 \hat{j}-5 \hat{k}\) and \(\vec{c}=-3 \hat{i}+2 \hat{j}+3 \hat{k}\), find the magnitude and direction cosines of
(i) \(\vec{a}+\vec{b}+\vec{c}\)
(ii) \(3 \vec{a}-2 \vec{b}+5 \vec{c}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 23
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 24>

Samacheer Kalvi 11th Maths Question 12.
The position vectors of the vertices of a triangle are Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 25 and Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 26. Find the perimeter of the triangle
Solution:
Let A, B, C be the vertices of the triangle ABC,
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 27

11 Samacheer Maths Solutions Question 13.
Find the unit vector parallel to Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 28 and Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 29
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 30
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 31

12th Maths Exercise 8.2 Samacheer Kalvi Question 14.
The position vector \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) three points satisfy the relation \(2 \vec{a}-7 \vec{b}+5 \vec{c}=\overrightarrow{0}\). Are these points collinear?
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 32

10th Maths Exercise 8.2 Samacheer Kalvi Question 15.
The position vectors of the point P, Q, R, S are Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 33 and Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 34 respectively. Prove that the line PQ and RS are parallel.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 35

Ex 8.2 Class 11 Question 16.
Find the value or values of m for which \(m(\hat{i}+\hat{j}+\hat{k})\) is a unit vector
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 36

11th Maths 8th Chapter Question 17.
Show that the points A(1, 1, 1), B(1, 2, 3) and C(2, -1, 1) are vertices of an isosceles triangle.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 37

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.2 Additional Problems

Samacheer Kalvi Maths Guide 11th Question 1.
Show that the points whose position vectors given by
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 38
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 39
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 40
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 41

Samacheer Kalvi.Guru 11th Maths Question 2.
Find the unit vectors parallel to the sum of \(3 \hat{i}-5 \hat{j}+8 \hat{k}\) and \(-2 \hat{j}-2 \hat{k}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 42

Maths Guide 11th Question 3.
The vertices of a triangle have position vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 43 Prove that the triangle is equilateral.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 44
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 45

Maths Solutions Class 11 Samacheer Kalvi Question 4.
Prove that the points Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 46 form an equilateral triangle.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 47

Samacheer Kalvi 11th Maths Solution Book Question 5.
Examine whether the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 48 are coplanar
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 49
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.2 50
⇒ We are not able to write one vector as a linear combination of the other two vectors
⇒ the given vectors are not coplanar.

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