Characteristics of Vector Math Addition

In Mathematics, a vector is a quantity that has both magnitude and direction. It is often represented by an arrow with the same direction as the amount and a length proportionate to the magnitude of the quantity. A vector, although having magnitude and direction, does not have a location. In other words, if a vector is shifted parallel to itself, its length remains unchanged. Scalars are ordinary quantities that have a magnitude but no direction, as opposed to vectors. Displacement, velocity, and acceleration, for example, are vector quantities, whereas speed (the magnitude of velocity), time, and mass are scalars.

Vector Algebra

Vector algebra is used to execute a variety of algebraic operations on vectors. A vector is a term from Latin that means “carrier.” Vectors connect points A and B. The length of the line connecting points A and B is known as the vector’s magnitude, and the direction or displacement from point A to point B is known as the vector’s direction AB. Vectors are sometimes referred to as Euclidean vectors and Spatial vectors. Vectors have several uses in mathematics, physics, engineering, and a variety of other subjects.

Addition of vectors

Adding two or more vectors is known as vector addition. When adding vectors, we use the addition operation to combine two or more vectors to create a new vector that is equal to the total of the vectors. Vector addition is utilized in physical quantities where velocity, displacement, and acceleration are represented by vectors.

Adding vectors is like adding scalars. To obtain the final value, the constituent components of the relevant vectors are added:

a + b = (a1i + b1 j + c1 k)+ (a2i + b2 j + c2 k) = (a1, b1, c1) + (a2, b2, c2) = (a1 + a2, b1 + b2, c1 + c2) = (a1 + a2) i + (b1 + b2) j + (c1 + c2) k

Vector addition is both commutative and associative.

Vector addition is governed by two laws

• Triangle law of vector addition: According to the law, if two sides of a triangle represent two vectors (both in magnitude and direction) operating in the same sequence on a body, then the third side of the triangle represents the resulting vector.

• Parallelogram law of vector addition: If two co-initial vectors functioning concurrently are represented by two adjacent sides of a parallelogram, then the diagonal of the parallelogram represents the sum of the two vectors, that is, the resulting vector beginning from the same initial point.

Subtraction of vectors

Vector subtraction works similarly to vector addition. However, in this case, just the sign of one of the vectors is altered and added to the other vector.

a – b = (a1 i + b1 j + c1k)- (a2i + b2j + c2 k) = (a1, b1, c1) – (a2, b2, c2) = (a1 – a2, b1 – b2, c1 – c2) = (a1 – a2) i + (b1 – b2) j + (c1 – c2) k

Characteristics of vector addition

• A vector can only be added to another vector.

• The addition closure property states that the sum of two vectors is always a vector. As a result, vectors are closed under addition.

• Vector addition is a commutative operation.

• Vector addition is an associative operation.

• Vector addition has a distributive property.

• The magnitude of the resultant of two vectors is less than or equal to the sum of the magnitudes of the two vectors and higher than or equal to the magnitude of the magnitude of the difference between the magnitudes of the two vectors.

Point to remember

• Vectors are shown with an arrow representation and are represented as a combination of direction and magnitude.

• We can determine the resultant vector if the components of a vector are provided.

• For the addition of vectors, the renowned triangle law can be utilized, and this method is also known as the head-to-tail method.

Conclusion

In mathematics, a vector is a geometric object that possesses both magnitude and direction. Vectors have an initial point where they begin and a terminal point that reveals the point’s end position. Vector algebra allows you to do algebraic operations like addition, subtraction, and multiplication. Many physical quantities, such as velocity, displacement, acceleration, and force, are vector values with both a magnitude and a direction.