Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.4

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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.4

Question 1.
Find the magnitude of Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 2

Question 2.
Show that Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 3
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 4

Question 3.
Find the vectors of magnitude \(10 \sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2 \hat{j}+\hat{k}\) and \(\hat{i}+3 \hat{j}+4 \hat{k}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 5
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 6

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4

Question 4.
Find the unit vectors perpendicular to each of the vectors
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 7
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 8

Question 5.
Find the area of the parallelogram whose two adjacent sides are determined by the vectors \(\hat{i}+2 \hat{j}+3 \hat{k}\) and \(3 \hat{i}-2 \hat{j}+\hat{k}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 9

Question 6.
Find the area of the triangle whose vertices are A(3, -1, 2), B(1, -1, -3) and C(4, -3, 1)
Solution:
A = (3, -1, 2); B = (1, -1, -3) and C = (4, -3, 1)
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 10

Question 7.
If \(\vec{a}, \vec{b}, \vec{c}\) are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is \(\frac{1}{2}|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|\). Also deduce the condition for collinearity of the points A, B, C
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 11
If the points A, B, C are collinear, then the area of ∆ABC = 0.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 12

Question 8.
For any vector \(\vec{a}\) prove that Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 13
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 14
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 15

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4

Question 9.
Let \(\vec{a}, \vec{b}, \vec{c}\) be unit vectors such that \(\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{b}}=\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{c}}=\mathbf{0}\) and the angle between \(\vec{b} \text { and } \vec{c} \text { is } \frac{\pi}{3}\). Prove that \(\vec{a}=\pm \frac{2}{\sqrt{3}}(\vec{b} \times \vec{c})\)
Solution:
Given \(|\vec{a}|=|\vec{b}|=|\vec{c}|\) = 1
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 16

Question 10.
Find the angle between the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 17 using vector product
Solution:
The angle between \(\vec{a}\) and \(\vec{b}\) using vector product is given by
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 18

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.4 Additional Problems

Question 1.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 19
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 20

Question 2.
If \(\vec{a}\), \(\vec{b}\) are any two vectors, then prove that Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 21
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 22

Question 3.
Find the angle between the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 23 by using cross product.
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 24
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 25

Question 4.
Find the vector of magnitude 6 which are perpendicular to both the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 26
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 27

Question 5.
Find the vectors whose length 5 which are perpendicular to the vectors Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 28
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 29

Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4

Question 6.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 30
Solution:
Given \(\vec{a} \times \vec{b}=\vec{c} \times \vec{d}\) and \(\vec{a} \times \vec{c}=\vec{b} \times \vec{d}\)
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 31

Question 7.
Find the angle between two vectors \(\vec{a}\) and \(\vec{b}\) if \(|\overrightarrow{\boldsymbol{a}} \times \overrightarrow{\boldsymbol{b}}|=\overrightarrow{\boldsymbol{a}} \cdot \overrightarrow{\boldsymbol{b}}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 32

Question 8.
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 33
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 34

Question 9.
If Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 35 find the angle between \(\vec{a}\) and \(\vec{b}\)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra - I Ex 8.4 36