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Vectors And Their Properties

Class 12th Mathematics RS Aggarwal Solution
Exercise 22
  1. Write down the magnitude of each of the following vectors:A. vector {a} =…
  2. Find a unit vector in the direction of the vector:A. ( 3 {i}+4 hat{j}-5…
  3. If vector {a} = ( 2 {i}-4 hat{j}+5 hat{k} ) then find the value of λ so…
  4. If vector {a} = ( - {i} + hat{j} - hat{k} ) and vector {b} = ( 2 {i} -…
  5. If vector {a} = ( 3 {i} + hat{j}-5 hat{k} ) and vector {b} = ( {i}+2…
  6. If vector {a} = ( {i}+2 hat{j}-3 hat{k} ) and vector {b} = ( 2 {i}+4…
  7. Find a vector of magnitude 9 units in the direction of the vector ( - 2 {i}…
  8. Find a vector of magnitude 8 units in the direction of the vector ( 5 {i} -…
  9. Find a vector of magnitude 21 units in the direction of the vector ( 2 {i}-3…
  10. If vector {a} = ( {i}-2 hat{j} ) , vec{b} = ( 2 hat{i}-3 hat{j} ) and…
  11. If A(-2, 1, 2) and B(2, -1, 6) are two given points, find a unit vector in the…
  12. Find the direction ratios and direction cosines of the vector vector {a} = (…
  13. Find the direction ratios and the direction cosines of the vector joining the…
  14. Show that the points A, B and C having position vectors ( {i}+2 hat{j}+7…
  15. The position vectors of the points A, B and C are ( 2 {i} + hat{j} - hat{k}…
  16. If the position vectors of the vertices A, B and C of a ∆ABC be ( {i}+2…
  17. Show that the points A, B and C having position vectors ( 3 {i}-4 hat{j}-4…
  18. Using vector method, show that the points A(1, -1, 0), B(4, -3, 1) and C(2,…
  19. Find the position vector of the point which divides the join of the points (…
  20. The position vectors of two points A and B are ( 2 vector {a} + vec{b} )…
  21. Find the position vector of a point R which divides the line joining A(-2, 1,…
  22. Find the position vector of the mid-point of the vector joining the points a…
  23. If vector {ab} = ( 2 {i} + hat{j}-3 hat{k} ) and A(1, 2, -1) is the given…
  24. Write a unit vector in the direction of vector {pq} where P and Q are the…
Exercise 22
Question 1.

Write down the magnitude of each of the following vectors:

A.

B.

C.

D.


Answer:

Tip – For any vector the magnitude


A.




B.




C.




D.





Question 2.

Find a unit vector in the direction of the vector:

A.

B.

C.

D.


Answer:

Tip – For any vector the unit vector is represented as


A.




B.




C.




D.





Question 3.

If then find the value of λ so that may be a unit vector.
Answer:



For a unit vector, its magnitude equals to 1.


Tip – For any vector the magnitude







Question 4.

If and then find the unit vector in the direction of


Answer:






Tip – For any vector the unit vector is represented as






Question 5.

If and then find a unit vector in the direction of


Answer:






Tip – For any vector the unit vector is represented as






Question 6.

If and then find a unit vector parallel to


Answer:






Tip – For any vector the unit vector is represented as







Question 7.

Find a vector of magnitude 9 units in the direction of the vector


Answer:

Let λ be an arbitrary constant and the required vector is


Tip – For any vector the magnitude





The required vector is



Question 8.

Find a vector of magnitude 8 units in the direction of the vector


Answer:

Let λ be an arbitrary constant and the required vector is


Tip – For any vector the magnitude





The required vector is



Question 9.

Find a vector of magnitude 21 units in the direction of the vector


Answer:

Let λ be an arbitrary constant and the required vector is


Tip – For any vector the magnitude





The required vector is



Question 10.

If and find


Answer:








Question 11.

If A(-2, 1, 2) and B(2, -1, 6) are two given points, find a unit vector in the direction of


Answer:

A = (-2, 1, 2)


B = (2, -1, 6)





Tip – For any vector the unit vector is represented as







Question 12.

Find the direction ratios and direction cosines of the vector


Answer:


Tip – For any vector the direction ratios are represented as (ax , ay ,az) and the direction cosines are given by


The direction ratios are (5,-3, 4)


The direction cosines are





Question 13.

Find the direction ratios and the direction cosines of the vector joining the points A(2, 1, -2) and B(3, 5, -4).
Answer:

A = (2,1,-2)


B = (3,5,-4)





Tip – For any vector the direction ratios are represented as (ax , ay ,az) and the direction cosines are given by


The direction ratios are (1,4, -2)


The direction cosines are




Question 14.

Show that the points A, B and C having position vectors and respectively, are collinear.
Answer:











So, the points A, B and C are collinear.
Question 15.

The position vectors of the points A, B and C are and respectively. Show that the points A, B and C are collinear.
Answer:











So, the points A, B and C are collinear.
Question 16.

If the position vectors of the vertices A, B and C of a ∆ABC be and respectively, prove that ∆ABC is equilateral.
Answer:













Tip – For any vector the magnitude






The three sides of the triangle are equal in magnitude, so the triangle is equilateral.
Question 17.

Show that the points A, B and C having position vectors and respectively, form the vertices of a right-angled triangle.
Answer:













Tip – For any 2 perpendicular vectors & ,






The triangle is right-angled.
Question 18.

Using vector method, show that the points A(1, -1, 0), B(4, -3, 1) and C(2, -4, 5) are the vertices of a right-angled triangle.
Answer:

A = (1,-1,0)


B = (4,-3,1)


C = (2,-4,5)











Tip – For any 2 perpendicular vectors & ,






The triangle is right-angled.
Question 19.

Find the position vector of the point which divides the join of the points and (i) internally and (ii) externally in the ratio 2 : 3.
Answer:



Formula to be used – The point dividing a line joining points a and b in a ratio m:n internally or externally is given by respectively.


The position vector of the point dividing the line internally




The position vector of the point dividing the line externally
Question 20.

The position vectors of two points A and B are and respectively. Find the position vector of a point C which divides AB externally in the ratio 1 : 2. Also, show that A is the mid-point of the line segment CB.
Answer:



Formula to be used – The point dividing a line joining points a and b in a ratio m:n externally is given by respectively.


The position vector of the point C dividing the line externally




The midpoint of B and C may be given by



i.e. point A


A is the midpoint of B and C.
Question 21.

Find the position vector of a point R which divides the line joining A(-2, 1, 3) and B(3, 5, -2) in the ratio 2 : 1 (i) internally (ii) externally.
Answer:

A = (-2,1,3)


B = (3,5,-2)




Formula to be used – The point dividing a line joining points a and b in a ratio m:n internally or externally is given by respectively.


The position vector of the point dividing the line internally




The position vector of the point dividing the line externally
Question 22.

Find the position vector of the mid-point of the vector joining the points and


Answer:



Formula to be used – The midpoint of a line joining points a and b is given by .


The position vector of the midpoint
Question 23.

If and A(1, 2, -1) is the given point, find the coordinates of B.
Answer:

A = (1,2,-1)


Let the co-ordinates of point B be (b1,b2,b3)




Comparing the respective co-efficient,


b1-1 = 2 i.e. b1 = 3


b2-2 = 1 i.e. b2 = 3


b3+1 = -3 i.e. b3 = -4


The required co-ordinates of B are (3,3,-4)
Question 24.

Write a unit vector in the direction of where P and Q are the points (1, 3, 0) and (4, 5, 6) respectively.
Answer:

P = (1,3,0)


Q = (4,5,6)





Tip – For any vector the unit vector is represented as