Maharashtra Board Text books

Chapter 5 Vectors Ex 5.2

Question 1.

word image 19203 2

(ii) externally.
Solution:
If R(r¯) divides the line segment joining P and Q externally in the ratio 3 : 2, by section formula for external division,
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 2
∴ coordinates of R = (-19, 8, -21).
word image 19203 4

Question 2.
Find the position vector of midpoint M joining the points L (7, -6, 12) and N (5, 4, -2).
Solution:

word image 19203 6
∴ coordinates of M = (6, -1, 5).
word image 19203 7

Question 3.
If the points A(3, 0, p), B (-1, q, 3) and C(-3, 3, 0) are collinear, then find
(i) The ratio in which the point C divides the line segment AB.
Solution:

word image 19203 8
As the points A, B, C are collinear, suppose the point C divides line segment AB in the ratio λ : 1.
∴ by the section formula,
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 4
By equality of vectors, we have,
-3(λ + 1) = -λ + 3 … (1)
3(λ + 1 ) = λ q … (2)
0 = 3λ + p … (3)
From equation (1), -3λ – 3 = -λ + 3
∴ -2λ = 6 ∴ λ = -3
∴ C divides segment AB externally in the ratio 3 : 1.

(ii) The values of p and q.
Solution:

Putting λ = -3 in equation (2), we get
3(-3 + 1) = -3q
∴ -6 = -3q ∴ q = 2
Also, putting λ = -3 in equation (3), we get
0 = -9 + p ∴ p = 9
Hence p = 9 and q = 2.

Question 4.

word image 19203 11
Hence, the position vector of C is 3a¯ – b¯.

Question 5.
Prove that the line segments joining mid-point of adjacent sides of a quadrilateral form a parallelogram.
Solution:

Let ABCD be a quadrilateral and P, Q, R, S be the midpoints of the sides AB, BC, CD and DA respectively.
word image 19203 12
Since P, Q, R and S are the midpoints of the sides AB, BC, CD and DA respectively,
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 6
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 7
∴ □PQRS is a parallelogram.

Question 6.
D and E divide sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE.
Solution:

Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 19
Let AD and BE intersect at P.
word image 19203 16
D and E divide segments BC and CA internally in the ratio 2 : 3.
By the section formula for internal division,
word image 19203 17
LHS is the position vector of the point which divides segment AD internally in the ratio 15 : 4.
RHS is the position vector of the point which divides segment BE internally in the ratio 10 : 9.
But P is the point of intersection of AD and BE.
∴ P divides AD internally in the ratio 15 : 4 and P divides BE internally in the ratio 10 : 9.
Hence, the position vector of the point of interaction of
word image 19203 18

Question 7.
Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.
Solution:

Let a¯, b¯, c¯ and d¯ be respectively the position vectors of the vertices A, B, C and D of the parallelogram ABCD. Then AB = DC and side AB || side DC.
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 10
word image 19203 20

Question 8.
Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half the sum of parallel sides.
Solution:

word image 19203 21
word image 19203 22
word image 19203 23
Then seg MN is the median of the trapezium.
By the midpoint formula,
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 12
word image 19203 25
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 13

Question 9.
If two of the vertices of the triangle are A(3, 1, 4) and B(-4, 5, -3) and the centroid of a triangle is G(-1, 2, 1), then find the coordinates of the third vertex C of the triangle.
Solution:
word image 19203 27
word image 19203 28
∴ the coordinates of third vertex C are (-2, 0, 2).

Question 10.
In ∆OAB, E is the mid-point of OB and D is the point on AB such that AD : DB = 2 : 1.
If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.
Solution:

Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 15
word image 19203 30
∵ AD : DB = 2 : 1.
∴ D divides AB internally in the ratio 2 : 1.
Using section formula for internal division, we get
word image 19203 31
LHS is the position vector of the point which divides OD internally in the ratio 3 : 2.
RHS is the position vector of the point which divides AE internally in the ratio 4 : 1.
But OD and AE intersect at P
∴ P divides OD internally in the ratio 3 : 2.
Hence, OP : PD = 3 : 2.

Question 11.
If the centroid of a tetrahedron OABC is (1, 2, -1) where A = (a, 2, 3), B = (1, b, 2), C = (2, 1, c) respectively, find the distance of P (a, b, c) from the origin.
Solution:

Let G = (1, 2, -1) be the centroid of the tetrahedron OABC.
word image 19203 32
Maharashtra Board 12th Maths Solutions Chapter 5 Vectors Ex 5.2 17
By equality of vectors
a + 3 = 4, b + 3 = 8, c + 5= -4
∴ a = 1, b = 5, c = -9
∴ P = (a, b, c) = (1, 5, -9)
word image 19203 34

Question 12.
Find the centroid of tetrahedron with vertices K(5, -7, 0), L(1, 5, 3), M(4, -6, 3), N(6, -4, 2) ?
Solution:

word image 19203 35
word image 19203 36
Hence, the centroid of the tetrahedron is G = (4, -3, 2).

Scroll to Top