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

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

Question 1.
Find $$\vec{a} \cdot \vec{b}$$ when Solution: Question 2.
Find the value of λ for which the vectors $$\vec{a}$$ and $$\vec{b}$$ are perpendicular, where Solution:
(i) $$\vec{a}$$ = 2î + λĵ – k̂ and $$\vec{b}$$ = î – 2ĵ + 3k̂
Given $$\vec{a}$$ and $$\vec{b}$$ are perpendicular vectors
∴ $$\vec{a}$$ . $$\vec{b}$$ = 0
(2î + λĵ – k̂ ) . (î – 2ĵ + 3k̂) = 0
(2) (1) + (λ) (- 2) + (1) (3) = 0
2 – 2λ + 3 = 0
2λ = 5
λ = $$\frac{5}{2}$$

(ii) $$\vec{a}$$ = 2î + 4ĵ – k̂ and $$\vec{b}$$ = 3î – 2ĵ + λk̂
Given $$\vec{a}$$ and $$\vec{b}$$ are perpendicular vectors
∴ $$\vec{a}$$ . $$\vec{b}$$ = 0
(2î + 4ĵ – k̂) . (3î – 2ĵ + λk̂) = 0
(2) (3) + (4) (-2) + (-1) (λ) = 0
6 – 8 – λ = 0
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:  Question 4.
Find the angle between the vectors Solution: 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: Question 6.
Show that the vectors are mutually orthogonal.
Solution: Question 7.
Show that the vectors form a right-angled triangle.
Solution: So, the given vectors form the sides of a right-angled triangle

Question 8. Solution: 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  Question 10.
If $$\vec{a}, \vec{b}$$ are unit vectors and θ is the angle between them, show that Solution: 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: 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: Question 13.
Find λ, when the projection of is 5 units.
Solution: Question 14. Solution:  ### Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.3 Additional Problems

Question 1.
Find λ so that the vectors are perpendicular to each other.
Solution:  Question 2. Solution:  Question 3.
If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is $$\sqrt{3}$$
Solution: Question 4.
Show that the vectors form a right-angled triangle.
Solution: ⇒ The given vectors form the sides of a right-angled triangle. Question 5.
Find the projection of Solution:  Question 6.
Show that the vector $$\hat{i}+\hat{j}+\hat{k}$$ is equally inclined with the coordinate axes.
Solution: 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: Question 8.
Show that the points whose positions vectors from a right-angled triangle.
Solution: ⇒ The given points form a right-angled triangle.

Question 9. Solution: 