Vector Operations

A vector is an object with both magnitude and direction. A vector can be visualized geometrically as a guided line segment with an arrow indicating the direction and a length equal to the magnitude of the vector. If two vectors have the same magnitude and direction, they are the same.

Vector Operations

Vector algebra refers to the algebraic operations in vector calculus that are defined for a vector space and then applied globally to a vector field. The following are the basic vector algebraic operations:

• Vector addition

• Vector subtraction

• Scalar multiplication

• Dot product

• Cross product

The vector calculation rules of various vector operations are discussed below.

Vector Addition

The resultant vector can be computed by adding two vectors together. Vector addition is the method of combining two or more vectors.

Assume that a and b are not inherently identical vectors and that their magnitudes and directions can vary. The resultant of a and b equals 

a + b = (a1 + b1) e1 + (a2 + b2) e2 +(a3 + b3) e3  

Place the tail of the arrow b at the head of the arrow a, and then draw an arrow from the tail of a to the head of b to show the addition graphically. The vector a + b is represented by the new arrow drawn, as shown below.

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Because a and b are the sides of a parallelogram, and a + b is one of the diagonals, this vector addition method is sometimes referred to as the parallelogram rule. This point would also be the base point of a + b if a and b are connected vectors with the same base point. 

Associative Law of Vector Addition

The associative law of vectors states that regardless of the order or grouping in which vectors are grouped, the number of vectors remains the same.

To prove vector addition is associative consider three vectors \[\bar{A}\], \[\bar{B}\] and \[\bar{C}\] . 

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To obtain the resultant vector apply head to tail rule that is (\[\bar{A}\] + \[\bar{B}\]) and (\[\bar{B}\] + \[\bar{C}\])

From the diagram, the resultant vector will be 

\[\bar{OR}\] = \[\bar{OP}\] + \[\bar{PR}\]

\[\bar{R}\] = \[\bar{A}\] + (\[\bar{B}\] + \[\bar{C}\]) ………….(1)

and

\[\bar{OR}\] = \[\bar{OQ}\] + \[\bar{QR}\]

\[\bar{R}\] = (\[\bar{A}\] + \[\bar{B}\]) + \[\bar{C}\] ………….(2)

Now equating equations (1) and (2)

\[\bar{A}\] + (\[\bar{B}\] + \[\bar{C}\]) = (\[\bar{A}\] + \[\bar{B}\]) + \[\bar{C}\]

This is known as the associative law of vector addition.

Vector Subtraction

The difference of a vector can be computed by subtracting one vector from another. 

Assume that a and b are not inherently identical vectors and that their magnitudes and directions can vary. The difference of the vectors a and b is

a - b = (a1 - b1) e1 + (a2 - b2) e2 +(a3 - b3) e3  

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The geometric representation of subtracting two vectors is as follows: to subtract b from a, align the tails of a and b at the same location, and then draw an arrow from the head of b to the head of a. The vector (-b) + a is represented by this new arrow, with (-b) being the inverse of b and 

(-b) + a = a - b.

Vector Multiplication 

Vector multiplication is a term that refers to one of many methods for multiplying two or more vectors by themselves.

The various vector multiplication rules are as follows:

• Dot Product

The dot product, also known as the scalar product, is a mathematical operation that returns a scalar quantity from two vectors. The product of the magnitudes of the two vectors and the cosine of the angle between them is known as the dot product of two vectors. It is also known as the product of the first vector's projection onto the second vector and the magnitude of the second vector. The vector multiplication rules for the dot product is as follows:

A . B =|A| |B| Cos θ

• Cross Product

The cross product, also known as the vector product, is a binary operation that produces another vector from two vectors. The vector perpendicular to the plane determined by the cross product of two vectors in 3-space is defined as the vector whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. As a result, if n̂ is the perpendicular unit vector to the plane defined by vectors A and B as 

A x B =|A| |B| Sin θ n̂

• Triple Product

A triple product is a three-dimensional vector product, typically Euclidean vectors. The scalar-valued scalar triple product and, less often, the vector-valued vector triple product are both referred to as triple products.

• Scalar Triple Product

The scalar triple product is defined as the dot product of one vector with the cross product of the other two. It is also known as the mixed product, box product, or triple scalar product. Scalar triple product is given as 

a . (b x c) = det (a, b, c)

• Vector Triple Product

The cross product of one vector with the cross product of the other two is known as the vector triple product.

The vector triple product is given as follows:

a x (b x c) = (a . c)b - (a . b)c

Conclusion

The laws of elementary algebra are extended to vectors by vector operations. Addition, subtraction, and three forms of multiplication are among them. The diagonal of the parallelogram built with the two initial vectors as sides are the sum of two vectors, which is defined as a third vector. When a vector is multiplied by a positive scalar, its magnitude is multiplied by the scalar, and its direction is unchanged; however, if the scalar is negative, the direction is reversed. The dot product, written a b, and the cross product, written a b, are the results of multiplying a vector by another vector b. For vector addition and the dot product, the associative and commutative laws apply. The cross product is not commutative, but it is associative.