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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:

(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: