Subtraction

Vectors depict entities that have a magnitude and act in a specific direction. An arrow depicts them. The head of the arrow points in the direction in which the vector is acting. The length of the arrow is the magnitude of the vector. All vectors have points from which they begin and directions towards which they point. A vector can be depicted with a plus or minus sign regarding its direction in one dimension. In a two-dimensional space, however, the direction of the vector is given concerning some object or system, for example, the coordinate system. Examples of vector quantities are displacement, velocity, force, etc. It is crucial to understand how addition and subtraction are done when working with vectors.

Angle between vectors

The angle between the vectors is critical when subtraction of vectors is done. This is because the angle between the vectors determines the direction and magnitude of the resultant vector. So, the dot product can calculate the angle between two vectors. The formula for the dot product of two vectors a and b is a.b = |a|.|b|cosθ. Where θ is the angle between the two vectors. So, the angle between the vectors can be calculated by the formula θ = cos -1[a.b / |a|.|b|]

Types of vectors

There are several types of vectors. They are as follows:

• Position vectors: These vectors denote the direction and position of vectors moving in three-dimensional space. Position vectors can be related to other objects, and their magnitude and direction can change based on their relation to other objects

• Negative vector: If a vector has the same magnitude as another vector, but its direction is opposite, then the vector is negative of the other vector

• Co-initial vectors: Vectors that have the exact origin are called co-initial vectors

• Orthogonal vectors: If the angle between vectors is 90 degrees, they are orthogonal

• Parallel vector: if two vectors have different magnitudes but are inclined at the same angle in the same direction, they are known as parallel vectors. The difference between the angles of parallel vectors is zero. If two vectors have a different magnitude and the difference between their angles is 180 degrees, they are known as antiparallel. 

• Equal vectors: Vectors that have the same magnitude and direction are said to be equal vectors. The corresponding components of equal vectors are equal. Equal vectors can have different points of origin and termination

• Unit vectors: The magnitude of unit vectors is equal to 1. They are also known as the multiplicative identity of vectors. The primary purpose of unit vectors is to give the direction of the vectors

• Zero vector: A vector with no magnitude and direction is known as a zero vector. It is represented by = (0,0,0). These vectors are also called the additive identity of vectors.

Subtraction of vectors

The subtraction of vectors can be done by the graphical and analytical methods or the parallelogram and the triangle methods. And the subtraction of vectors must follow some rules, which are as follows:

• Subtraction of vectors can take place only between vectors.

• The vectors must be of the same physical quantity.

1. Graphical subtraction of vectors

To understand what it means to subtract vectors, it is crucial to know what subtraction of vectors means. When a vector B is subtracted from a vector A, it simply means that the negative vector B is added to vector A. This can be written in the form of an equation in the following way A + (-B). The negative of any vector has the same magnitude but its direction opposite the vector. So subtraction of vectors is an extension of the addition of vectors. It is the addition of a negative vector to another vector. It can be done graphically in the following way. 

• Draw an arrow representing the first vector

• Draw the second vector so that its tail is touching the head of the first vector

• Draw an arrow connecting the tail of the first vector to the head of the second vector

• Measure this vector with a ruler. This is the resultant vector

• The direction of the vector can be found by measuring the angle that the resultant vector makes with the reference frame.

2. Analytical subtraction of vectors

Analytical methods of vector subtraction have an advantage of accuracy over the graphical methods. This is because the graphical method of drawing and measuring the vectors is subject to the accuracy of the drawing. On the other hand, analytical methods are determined by how accurate the measurement of the pg=hysical quantities is done. The analytical method of vector subtraction uses geometry and components of simple trigonometry to find out the magnitude and direction of vectors. So a significant part of analytical methods of vector calculation is to find out the perpendicular components of vectors. Perpendicular components are the perpendicular vectors that result in the vector under advisement when added. The Pythagorean theorem relates the perpendicular components of the vector and the vector itself. Supposing there is a vector A and its perpendicular components are Ax and Ay, then A = √[(Ax)2 + (Ay)2]

Suppose there are two vectors, A and B. Vector B is subtracted from vector A. The resultant vector can be calculated in the following way:

• Find out the individual perpendicular components of the A and B vectors. Let us assume they are Ax, Ay, Bx, By.

• Since vector B has to be subtracted, its perpendicular components will assume a negative value

• The above components are added together to find out the perpendicular components of the resultant vector R. So Rx= Ax – Bx and Ry = Ay –  By

• Now that the perpendicular components of the resultant vector are known, the resultant vector can be calculated using the Pythagorean theorem.

3. The parallelogram method

Subtraction of vectors is done by this method in the following way:

• Draw the two vectors with their initial points coinciding to become co-initial vectors

• Invert the vector B so that its direction becomes opposite. It is now a negative vector of B to be written as -B

• Complete the parallelogram

• Draw a diagonal from the initial point of the two vectors to the opposite vertex.

• This diagonal is the resultant vector.

4. The triangle method

This method can be used to subtract one vector from another in the following steps-

• Draw the two vectors, so they are co-initial vectors

• Join their ends by a straight line

• This straight line is the resultant vector

• The tail of the resultant vector is always towards the head of the negative vector in this method.

It is vital to have well-elaborated vector subtraction study material if one aims to perform calculations involving vectors.

Conclusion

Subtraction of vectors is an integral part of working with vectors. It is used in many situations where calculations dealing with force, displacement, acceleration, and velocity must be done. The methods used to subtract vectors are graphical, analytical, parallelogram, and triangle. However, the analytical method that employs trigonometry and geometry is the most accurate. The parallelogram method and the triangle methods are derivative of the analytical method since principles of geometry are used to determine the resultant vector. Having comprehensible study material notes on subtraction of vectors is necessary to understand working with vectors in problems or mechanics.