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

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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:
Given (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.
Let the given points be A (1, 0, 0) and B (0, 1, 0)
[The direction ratios of the line joining the points A (x₁, y₁, z₁) and B (x₂, y₂, z₂) are x₂ – x₁, y₂ – y₁, z₂ – z₁ or x₁ – x₂, y₂ – y₂, z1,  z₁ – z₂ j
The direction ratios of the line joining the points A(1, 0, 0) and B(0, 1, 0) are
(0 – 1, 1 – 0, 0 – 0)
(- 1, 1, 0) ………. (1)
Also the direction ratios are
(1 – 0, 0 – 1, 0 – 0)
(1, -1, 0) ………. (2)
Given the direction ratios are
(a, a + b , a + b + c) ………. (3)
Comparing (1) and (3) we have
(-1, 1, 0) = (a, a + b, a + b + c)
a = -1, a + b = 1 ⇒ – 1 + b = 1 ⇒ b = 2
a + b + c = 0 ⇒ – 1 + 2 + c = 0 ⇒ c = – 1
Comparing (2) and (3) we get
a = 1 , a + b = – 1 ⇒ 1 + b = – 1 ⇒ b = – 2
a + b + c = 0 ⇒ 1 – 2 + c = 0 ⇒ c = 1
∴ The required set of values of a, b, c are
a = -1, b = 2, c = – 1 and
a = 1, b = – 2, 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:

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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 points 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 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.