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.

Table of Content

• Negative of a Bector
• Graphical Subtraction of Vectors
• Mathematical Formulation of Subtraction
• Properties of Subtraction of Vector
• Practice Problems of Vector Subtraction
• FAQs of Vector Subtraction

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 , then be the negative of vector . and can be written as

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 and then subtraction of these vector is represented by  

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

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

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

Mathematical Formulation of Subtraction

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

Properties of Subtraction of Vector

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

• Similarly, subtracting vectors is not associative.

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Practice Problems of Vector Subtraction

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

Answer: Velocity of A with respect to                                                           

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   , and the second leg of the trip with a vector .

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 . Therefore, he will end up at a location or .

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

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

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

Answer: From the subtraction of vector we know,

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

Answer: Angle

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 with .

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

• Scalar and Vector
• Types of Vector
• Lami's Theorem
• Vector Addition

NCERT Class 11 Physics Chapters