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Vector Subtraction: Methods, Mathematical Formulation, Properties

When you swim perpendicular to the flow in a river from one bank to another bank then you will observe that you're not going the path of shortest distance. It is because of the relative velocity between you and the flow of the river.

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Negative of a Vector

A vector is said to be the negative of a given vector if it has the same magnitude and opposite in direction.

Suppose we have a given vector IMAGE , then IMAGE be the negative of vector IMAGE. and can be written as

IMAGE

Here we can see both the vectors have the same magnitude but opposite in direction.

Subtraction of Vectors

Vector subtraction is the operation of finding the relative quantity of a vector with respect to other vector. 

Suppose we have two vector IMAGE and IMAGE then subtraction of these vector is represented by IMAGE 

IMAGE 

Also, we can define subtraction of two vector in terms of addition of two vector as,

IMAGE

Subtraction of two vectors is the addition of a vector IMAGE with the negative of vector IMAGE

From figure we can see negative of IMAGE is IMAGE, And resultant (Addition) of IMAGE and IMAGE is represented by IMAGE.

Mathematical Formulation of Subtraction

If IMAGE is the angle between vector IMAGE and IMAGE then IMAGE will be the angle between IMAGE and IMAGE then magnitude of vector IMAGE can be find as IMAGE

IMAGE

Properties of Subtraction of Vector

  • Subtracting vectors is not commutative. This is because vector IMAGE  are not the same (most of the time) and a negative sign affects a vector's direction.

IMAGE

  • Similarly, subtracting vectors is not associative.

IMAGE.

  • IMAGE

Practice Problems of Vector Subtraction

Question 1. Particle A is moving with the velocity IMAGE and particle B is moving with velocity IMAGE find the velocity of A with respect to B. 

Answer: Velocity of A with respect to IMAGE                                                          IMAGE

Question 2. A boy is sailing a boat at night. He first sail 27.5 M in a direction 660north of east from his current location, and then travel 30.0 m in a direction 1120 north of east. If the boy makes a mistake and travels in the opposite direction for the second leg of the trip, where will he end up?

Answer: We can represent the first leg of the trip with a vector IMAGE , and the second leg of the trip with a vector IMAGE.

Since the boy by mistake travels in the opposite direction for the second leg of the journey, the vector for the second leg of the trip will be IMAGE. Therefore, he will end up at a location IMAGE or IMAGE.

Note that IMAGE is of magnitude as B (30. 0 m), but in opposite direction, IMAGE south of east.

Draw the vectors IMAGE and IMAGE and the resultant. Use ruler and protractor find  magnitude and direction of resultant vector IMAGE

Question 3. Magnitude of IMAGE is 9 and IMAGE is 12 and angle between them is 600.find the IMAGE.

Answer: From the subtraction of vector we know,IMAGE

Question 4. If Magnitude of IMAGE is 7 and IMAGE is 5 and angle between them is 300.find the angle between IMAGE and vector IMAGE.

Answer: Angle

IMAGE

FAQs of Vector Subtraction

Question 1. What is negative of a vector?

Answer: Vector having the same magnitude and opposite in direction of a given vector is known as negative of a vector.

Question 2. What is the subtraction of a vector?

Answer: In general, It is addition of negative of vector IMAGE with IMAGE.

Question 3. What is the procedure of subtraction two vectors?

Answer: Keep two vectors tail to tail, now draw the resultant vector such that its head should be on the vector from which you are subtracting and tail on the vector which is subtracted.

Question 4. Is vector subtraction commutative.

Answer: No.

Related Topics to Vector Subtraction in Physics

NCERT Class 11 Physics Chapters

Physical World Units and Measurements

Motion in a Straight Line

Motion in a Plane Laws of Motion Work Energy and Power

Particles and Rotational Motion

Gravitation Mechanical Properties of Solids
Mechanical Properties in Liquids Thermal Properties of Matter Thermodynamics
Kinetic Theory Oscillations Waves
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