Content Curator
Displacement vector is a straight line from the starting point to the endpoint.
The displacement vector means how far has the object/person moved from the origin which can be found by drawing a straight line from the point of the start of the journey to the point from where one needs the displacement point. The displacement vector is always equal to or greater than the distance traveled as the distance is the length that has been covered while moving between two points while displacement is the straight line.
Scalars and Vectors
In Physics, quantities can be classified as scalars and vectors. Both have magnitude but only vectors have magnitude and direction. A scalar quantity only has magnitude. So, a scalar can be identified as a quantifiable number with a unit associated with it. For example Distance, Mass, Temperature, Time, etc. Ordinary algebra can be used to perform operations on scalar quantities.
Vectors having both magnitude and direction cannot be operated on using ordinary algebra. They obey the triangle law of addition or its equivalent, the parallelogram law of addition. For example Displacement, velocity, momentum, etc.
A vector can be represented as v or.
The magnitude of a vector is represented as | v | = v.
The body started from the origin O and traveled to C, then B, then A, and then back to O. The distance traveled will be according to the lengths of OC, CB, BA, and AO. But, the displacement of the body is zero. As the body has reached back to the point from where it started and there is no displacement i.e no difference in starting and ending point.
Also Read: Resolution of Vectors
Position Vector
The position vector represents the current position of an object/person at a particular time T relative to the starting point/origin. The motion of a body can be predicted or calculated according to the position vector.
The Position Vector can be represented as:
is the unit vector along the x-axis
is the unit vector along the y-axis
is the unit vector along the z-axis
Displacement Vector
When the object/body changes its position from point A to point B then this change in positions can be identified as a displacement vector.
In the figure, OA and OB are 2 position vectors while AB is the displacement vector.
Suppose in the above figure, the object was at point A at time t0 and at point B at time t1.
The position vectors for the object are given as:
Position Vector at Point A:
\(\hat r_A = 5 \hat i + 3 \hat j + 4 \hat k\)
Position Vector at Point B:
\(\hat r_B = 2 \hat i + 2 \hat j + 1 \hat k\)
Displacement Vector will be B - A:
\(\hat r_B - \hat r _A = - 3\hat i - \hat j - 3\hat k\)
Also Read: Unit Vectors
Things to Remember
- The magnitude of displacement is either less or equal to the path length of an object between two points.
- Vectors can have negative values but magnitudes are always positive. Therefore, scalars are always positive, while vectors can be positive/negative.
- Displacement can be zero if the object is back at the origin but distance can never be zero for an object which has moved.
- Vector operations are different from ordinary algebraic operations. They follow the triangle law of addition or parallelogram law of addition which calculates not only according to the magnitude but also the direction associated with a quantity.
- The displacement vector is always a straight line from the starting point to the current position of the object because a straight line is always the shortest distance between two points.
Sample Questions
Ques 1. A person goes 8 m north then 6 m east and 10 m vertically upwards. What is the displacement from the origin? (2 Marks)
Ans. The east direction is the x-axis, the north direction is the y-axis, and upwards is the z-axis. The displacement can be found by finding the magnitude of the resultant vector:
|| = m.
Ques 2. A dog is walking towards the well to drink water. It takes 5 steps forward and 2 steps backward. Each step is 1 m long and takes 1 second. The well is 11 m away from the origin. After how many seconds will the dog be able to reach the well? (3 Marks)
Ans. The dog is walking 5 steps forward and 3 steps backward in 8 seconds after which it is 2m away from an initial position.
Distance covered = 2 m in 8 seconds ( 5 + 3 )
After 24 seconds, the dog will be 6 m away from the starting position.
The dog will now move 5 steps forward to find the well which will take him 5 seconds.
Therefore, the total time taken by the dog to find the well is 24 + 5 = 29 seconds.
Ques 3. A man walks 6 m towards the east direction and then 8 m towards the north direction. What is the magnitude of displacement of the man? (2 Marks)
Ans.
The man has the following vector representation D =
So, to calculate the magnitude of the man’s displacement the vector’s magnitude has to be calculated as || = = = 10 m
Ques 4. Differentiate between distance and displacement. (5 Marks)
Ans.
| Distance | Displacement |
|---|---|
| The total path covered between two points without the consideration of the direction | The shortest distance between two points |
| Direction is not considered | Direction is considered |
| scalar quantity | vector quantity |
| speed x time | velocity x time |
| depends on the path taken | does not depend on the path. The only initial and final position |
| cannot be zero or negative | can be zero or negative |
| the distance cannot decrease with time | displacement can decrease with time |
Ques 5. Distinguish between position vector and displacement vector. (3 Marks)
Ans.
| Position Vector | Displacement Vector |
|---|---|
|
|
|
| It represents position | It represents a change in position vector |
| Fixed vector | Not fixed vector |
Ques 6. A = 4i + 3j and B = 5i + 4j. Find the resultant vector from the addition of these two vectors. (1 Mark)
Ans. Given 2 vectors, A = 4i + 3j and B = 5i + 4j
Resultant vector after addition of A and B →
= 4i + 3j + 5i + 4j = 9i + 7j
Ques 7. The position vector of the particle moving in a plane is given by, r = t2 + 3t. Find the displacement between t =1 and t = 4 seconds. (2 Marks)
Ans. r = t2 + 3t .
At time, t = 1 → ri = + 3
At time, t = 4 → rf = 16 + 12
The displacement between these position vectors will be given by the difference of their position vectors.
r = rf – ri
⇒ r = 16+ 12 -( + 3)
⇒ r = 15+ 9
Ques 8. Explain why displacement is a vector quantity? (2 Marks)
Ans. A vector quantity is defined as the physical quantity that has both directions as well as magnitude. Displacement is a vector quantity because it has both magnitude and direction.
The shortest distance between the initial and final point of an object within a particular time is considered to be displacement. It is defined as the change in the position of an object. Displacement is known as a vector quantity having a direction and magnitude. It is represented as an arrow that points from the starting position to the final position.
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