Scalar Multiplication of Vector Concepts Explained

Similar to other mathematical operations, multiplication of vectors results and a specific value. It is an operation that defines a vector space where a positive real number is multiplied by the magnitude of the vector. However, this multiplication does not result in any change in its direction.

Although scalar multiplication of vectors can be seen as a geometric interpretation of the main concept, it produces the vector in the same or opposite direction with a different length. As an exception, one of the factors may be taken as its original value, and scalar multiplication may be performed to test its field’s operation.

What is a vector? 

A vector is a quantity with direction and magnitude but no position in mathematics. The two most common examples of vectors are acceleration and velocity. It is represented as a line segment with length identified as its magnitude. Before we address our question – What is Scalar Multiplication of Vectors, let us first understand the conceptual meaning of scalar.

What is a scalar?

 A scalar is an element that defines a vector space. It is identified as a quantity described in terms of multiple scalars that possess both direction and magnitude, commonly known as vectors. In cases where the real numbers relate directly to the vector space through scalar multiplication, the scalar is defined to produce another vector.

In a few exceptional and applicable cases of scalar multiplication of vectors, the term is also used to refer to a matrix or vector that has reduced its value to a single component. For instance, the product of a 1 × m matrix and an m × 1 matrix can also be called a scalar.

What is scalar multiplication? 

Scalar multiplication of vectors refers to mathematical operations that define a vector space in linear algebra. The term is derived from its core usage, used for scales vectors. In other words, it is the multiplication of a vector and a scalar that is different from the inner product of two separate vectors.

From the geometrical perspective, scalar multiplication by a positive real number multiplies the vector’s magnitude from the Euclid vector. However, it does not bring any change in its direction.

Scalar multiplication of vectors can also be seen as an alternate binary operation where the field on the vector space identifies as a different operation. As per the geometric approach that defines the applicability of scalar multiplication, its vectors get stretched or contracted due to the participation of a constant factor. As a result, a vector is produced opposite the same direction of the primary vector. However, it has a different length.

Properties of scalar multiplication 

Let a and b be vectors, and c and d be scalars. The following properties can be held true for the scalar multiplication of vectors in this situation.

• The magnitude of the scaled vector will be equivalent to the absolute value. However, the value must relate to the scalar with respect to the magnitude of the vector. The resulting equation will be: ||cb|| = |c|b
• As per the associative property, when the two values are multiplied, the resulting value remains the same even if the two values are brought together differently. The resulting equation would be: c(da) = (cd)a
• As per the operation of commutative property, the operations on two values will change the order of their value in the original equation. However, it does not have any impact on the answer. Applying the same principle in our example, we get the following equation: ca = ac
• Following the applications of distributive property in our example of scalar multiplication of vectors, multiplying two values will result in a value obtained by multiplying each of them individually and then finding the sum of their products. The resulting equations would be: 

(c+d)a = ca + da

c(a+b) = ca + cb

• Applying the identity property, we get 1×a = a. After multiplying our value with one, it implies that the resulting number does not change the solution. In other words, the Identity of the value did not alter after multiplying it with 1.
• As per the results of the multiplicative property of -1, any value multiplied by -1 will result in a negative value. For instance, let us apply the property in our example. If we multiply -1 with c, the resulting value would be -c. Hence, it gives an additive inverse.
• According to the multiplicative property of zero (0), multiplying any value with 0 ultimately results in a 0. Applying the property in our example, 0(a) = 0, we get a zero vector for the scalar multiplication of vectors. 

Conclusion

In a nutshell, we can say that scalar multiplication of a vector refers to a mathematical operation whereby a vector is multiplied by a scalar, resulting in a distinct value defined as the inner product of two vectors. The operation also implies that various mathematical properties hold even in the case of scalar multiplication. These include additivity in the scalar and vector, compatibility of products of scalars, multiplication by zero to obtain zero vector, multiplying by -1 to give an additive inverse, and multiplying by 1, resulting in no change in the vector.