Negative of a Vector

Negative vectors have the same magnitude as their corresponding positive vectors, but they are directed in the opposite direction. It is obtained by multiplying the given vector by a factor of one hundred. The negative of a vector and the given vector are exactly in the same direction as each other.

Let us learn more about the negative of a vector by looking at a few situations that have been solved.

Negative of a Vector

a vector that has been multiplied by one (-1), or the negative of a vector, is called a vector. In other words, the vector -a is the inverse of the vector a. In this case, the magnitudes of a and -a are the same, but they are moving in different directions. i.e.,

|a| = |-a| 

a and -a are in diametrically opposed directions.

Assuming the vector AB is a vector from A to B, the negative vector of AB is BA, and it is a vector from B to A. In other words, just multiplying by a negative sign causes the vector to reverse direction. As a result, we can state that AB = – AB and that BA = = – AB. In addition, |AB| = |BA|. As a result, AB and BA are diametrically opposed to one another.

Finding the negative of a vector

To get the negative of any vector, simply multiply each of its components by a factor of -1. For example, if a = 1, -2, and -3, then -a = – a = 1, -2, and -3 = -1, -(-2), and -(-3)> = -1, 2, and 3 = -1, 2, and 3 In other words, to obtain the negative of a vector, we just need to reverse the signs of its constituent elements. In this scenario, the vectors a and -a are referred to as negative vectors of one another. Here are a few more illustrations.

Assuming v = x, -y, then -v = (-x, -y).

We already know that AB = OB – OA is true. As a result, BA = OA – OB is the negative vector.

The position vectors of points A and B are denoted by the letters OA and OB.

Magnitude of Negative of a Vector

The magnitude of a vector’s negative value

Knowing that the magnitude of any vector can never be negative is a given. It is always either a positive number or a zero. As a result, the magnitude of a negative vector is never equal to its negative sign. It is always equal to the magnitude of the real vector in the given situation. Here are some illustrations to help you comprehend what I’m talking about.

We have seen that if a = <1, -2, -3>, then -a = <-1, 2, 3>. Here,

|a| = √(1² + (-2)² + (-3)²) = √(1+4+9) = √14

|-a| = √((-1)² + 2² + 3²) = √(1+4+9) = √14

The fact that v=x,-y> follows from the fact that -v=x,-y> follows from the fact that Here, 

|v| = (x² + (-y)²) = (x² + y²)

 |v| = ((-x)² + y²) = (x² + y²) 

In each of these situations, the magnitude of the vector is the same as the magnitude of the vector’s negative counterpart.

The Characteristics of a Vector’s Negative

•Dot products are formed as follows: 

-a.-b = -1(a. b) 

•If a and B are cross products, then 

a x b = a x (-b) = -1(a x B).

•Additionally, axb = – (bxa). Therefore, the negative vectors of each other are denoted by the symbols ab and – (ba).

•The cross product of a vector and its negative vector is also a zero vector, as previously stated. This means that the product of , ax(-a)=-(axa)=0.

•A zero vector is formed by adding together a vector and its negative vector. Specifically, for each vector a, a + (-a) = 0.

Conclusion

A vector that has been multiplied by one (-1), or the negative of a vector, is called a vector. In other words, the vector -a is the inverse of the vector a. In this case, the magnitudes of a and -a are the same, but they are moving in different directions. i.e.,

|a| = |-a| ,a and -a are in diametrically opposed directions.To get the negative of any vector, simply multiply each of its components by a factor of -1.The cross product of a vector and its negative vector is also a zero vector, as previously stated. This means that the product of , ax(-a)=-(axa)=0.A zero vector is formed by adding together a vector and its negative vector. Specifically, for each vector a, a + (-a) = 0.