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

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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

Solution:

Question 2.
Find the value of λ for which the vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular, where

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:

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 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: