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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.3

Question 1.
Find \(\vec{a} \cdot \vec{b}\) when
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 2

Question 2.
Find the value of λ for which the vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular, where
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 3
Solution:
When \(\vec{a}\) and \(\vec{b}\) are ⊥r then \(\vec{a} \cdot \vec{b}\) = 0
\(\vec{a}\) ⊥r \(\vec{b}\) ⇒ \(\vec{a} \cdot \vec{b}\) = 0
(i) (2) (1) + (λ) (-2) + (1) (3) = 0 ⇒ λ = 5/2
(ii) (2) (3) + (4) (-2) + (-1) (λ) = 0
6 – 8 – λ = 0
-λ – 2 = 0 ⇒ -λ = 2 ⇒ λ = -2

Question 3.
If \(\vec{a}\) and \(\vec{b}\) are two vectors such that |\(\vec{a}\)| = 10, |\(\vec{b}\)| = 15 and \(\vec{a} \cdot \vec{b}\) = 75\(\sqrt{2}\) , find the angle between \(\vec{a}\) and \(\vec{a}\) .
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 4

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3

Question 4.
Find the angle between the vectors
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 5
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 6

Question 5.
If \(\overrightarrow{\boldsymbol{a}}, \overrightarrow{\boldsymbol{b}}, \overrightarrow{\boldsymbol{c}}\) are three vectors such that \(\vec{a}+2 \vec{b}+\vec{c}=\overrightarrow{0}\) and \(|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=7\) find the angle between \(\vec{a}\) and \(\vec{b}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 7

Question 6.
Show that the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 8 are mutually orthogonal.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 9

Question 7.
Show that the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 10 form a right angled triangle.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 11
So, the given vectors form the sides of a right angled triangle

Question 8.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 12
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 13

Question 9.
Show that the points (2, -1, 3) (4, 3, 1) and (3, 1, 2) are collinear
Solution:
Let the given points be A, B, C
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 14

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3

Question 10.
If \(\vec{a}, \vec{b}\) are unit vectors and θ is the angle between them, show that
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 15
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 16

Question 11.
Let \(\vec{a}, \vec{b}, \vec{c}\) be the three vectors such that \(|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5\) and each one of them being perpendicular to the sum of the other two, find \(|\vec{a}+\vec{b}+\vec{c}|\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 17

Question 12.
Find the projection of the vector \(\hat{i}+3 \hat{j}+7 \hat{k}\) on the vector \(2 \hat{i}+6 \hat{j}+3 \hat{k}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 18

Question 13.
Find λ, when the projection of Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 19 is 5 units.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 20

Question 14.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 21
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 22
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 23

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.3 Additional Problems

Question 1.
Find λ so that the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 24 are perpendicular to each other.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 25

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3

Question 2.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 26
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 27
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 28

Question 3.
If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is \(\sqrt{3}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 29

Question 4.
Show that the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 30 form a right angled triangle.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 31
⇒ The given vectors form the sides of a right of a right angled triangle.

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3

Question 5.
Find the projection of
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 32
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 33
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 34

Question 6.
Show that the vector \(\hat{i}+\hat{j}+\hat{k}\) is equally inclined with the coordinate axes.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 35

Question 7.
If \(\vec{a}, \vec{b}, \vec{c}\) are three mutually perpendicular unit vectors, then prove that \(|\vec{a}+\vec{b}+\vec{c}|=\sqrt{3}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 36

Question 8.
Show that the points whose positions vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 37 from a right angled triangle.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 38
⇒ The given points form a right angled triangle.

Question 9.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 39
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.3 40

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