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## Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.2

Question 1.

Verify whether the following ratios are direction cosines of some vector or not.

Solution:

Question 2.

Find the direction cosines of a vectors whose direction ratios are

(i) 1, 2, 3

(ii) 3, -1, 3

(iii) 0, 0, 7

Solution:

Question 3.

Find the direction cosines and direction ratios for the following vectors

Solution:

Question 4.

A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians.

Solution:

Question 5.

If \(\frac{1}{2}, \frac{1}{\sqrt{2}}\), a are the direction cosines of some vector, then find a.

Solution:

Question 6.

If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c.

Solution:

Let A be the point (1, 0, 0) and B be the point (0, 1, 0) (i.e.,) \(\overrightarrow{\mathrm{OA}}=\hat{i}\) and \(\overrightarrow{\mathrm{OB}}=\hat{j}\)

Then \(\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\hat{j}-\hat{i}=-\hat{i}+\hat{j}\)

= (-1, 1, 0)

= (a, a + b, a + b + c)

⇒ a = -1, a + b = 1 and a + b + c = 0

Now a = -1 ⇒ -1 + b = 1 ;a + b + c = 0

⇒ b = 2; -1 + 2 + c = 0 ⇒ c + 1 = 0

⇒ c = -1

∴ a = -1; b = 2; c = -1.

Note: If we taken \(\overrightarrow{\mathrm{BA}}\) then we get a = 1, b = -2 and c = 1.

Question 7.

Show that the vectors \(2 \hat{i}-\hat{j}+\hat{k}, 3 \hat{i}-4 \hat{j}-4 \hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}\) form a right angled triangle.

Sol:

⇒ The given vectors form the sides of a right angled triangle.

Question 8.

Find the value of k for which the vectors \(\vec{a}=3 \hat{i}+2 \hat{j}+9 \hat{k}\) and \(\vec{b}=\hat{i}+\lambda \hat{j}+3 \hat{k}\) are parallel.

Solution:

Question 9.

Show that the following vectors are coplanar.

Solution:

Let the given three vectors be \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\). When we are able to write one vector as a linear combination of the other two vectors, then the given vectors are called coplanar vectors.

We are able to write \(\vec{a}\) as a linear combination of \(\vec{b}\) and \(\vec{c}\)

∴ The vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are coplanar

Question 10.

Show that the points whose position vectors and are coplanar

Solution:

Let the given points be A, B, C and D. To prove that the points A, B, C, D are coplanar, we have to prove that the vectors \(\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}\) and \(\overrightarrow{\mathrm{AC}}\) are coplanar

∴ we are able to write one vector as a linear combination of the other two vectors ⇒ the given vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are coplanar.

(i.e.,) the given points A, B, C, D are coplanar.

Question 11.

If \(\vec{a}=2 \hat{i}+3 \hat{j}-4 \hat{k}\), \(\vec{b}=3 \hat{i}-4 \hat{j}-5 \hat{k}\) and \(\vec{c}=-3 \hat{i}+2 \hat{j}+3 \hat{k}\), find the magnitude and direction cosines of

(i) \(\vec{a}+\vec{b}+\vec{c}\)

(ii) \(3 \vec{a}-2 \vec{b}+5 \vec{c}\)

Solution:

>

Question 12.

The position vectors of the vertices of a triangle are and . Find the perimeter of the triangle

Solution:

Let A, B, C be the vertices of the triangle ABC,

Question 13.

Find the unit vector parallel to and

Solution:

Question 14.

The position vector \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) three points satisfy the relation \(2 \vec{a}-7 \vec{b}+5 \vec{c}=\overrightarrow{0}\). Are these points collinear?

Solution:

Question 15.

The position vectors of the point P, Q, R, S are and respectively. Prove that the line PQ and RS are parallel.

Solution:

Question 16.

Find the value or values of m for which \(m(\hat{i}+\hat{j}+\hat{k})\) is a unit vector

Solution:

Question 17.

Show that the points A(1, 1, 1), B(1, 2, 3) and C(2, -1, 1) are vertices of an isosceles triangle.

Solution:

### Samacheer Kalvi 11th Maths Solutions Chapter 8 Vector Algebra – I Ex 8.2 Additional Problems

Question 1.

Show that the points whose position vectors given by

Solution:

Question 2.

Find the unit vectors parallel to the sum of \(3 \hat{i}-5 \hat{j}+8 \hat{k}\) and \(-2 \hat{j}-2 \hat{k}\)

Solution:

Question 3.

The vertices of a triangle have position vectors Prove that the triangle is equilateral.

Solution:

Question 4.

Prove that the points form an equilateral triangle.

Solution:

Question 5.

Examine whether the vectors are coplanar

Solution:

⇒ We are not able to write one vector as a linear combination of the other two vectors

⇒ the given vectors are not coplanar.