Negative of a Vector

The negative of a vector is a vector with the same magnitude as the given vector but in the opposite direction. It is obtained by multiplying the given vector by -1. The negative of a vector and the given vector are exactly in the opposite directions.

Let us learn more about the negative of a vector along with a few solved examples.

The negative of a vector a is a vector that is obtained by multiplying it by -1. i.e., the negative of a is the vector -a. Here, a and -a are of same magnitudes but are in opposite directions. i.e.,

• |a| = |-a|
• a and -a have opposite directions.

If AB is a vector from A to B, then its negative vector is BA and it is from B to A. i.e., just multiplying by a negative sign changes the direction of the vector. In this case, we can say AB = - BA and BA = - AB. Also, |AB| = |BA|. Hence, AB and BA are the negatives of each other.

To find the negative of any vector, just multiply each of components by -1. For example, if a = <1, -2, -3>, then -a = - <1, -2, -3> = <-1, -(-2), -(-3)> = <-1, 2, 3>. i.e., to find the negative of a vector, we just need to change the signs of its components. In this case, a and -a are called negative vectors of each other. Here are more examples.

• If v = <x, -y> then -v = <-x, y>
• We know that AB = OB - OA. So its negative vector is BA = OA - OB.
  Here, OA and OB are the position vectors of points A and B.

We know that the magnitude of any vector is never negative. It is always either positive or 0. Thus, the magnitude of a negative of a vector is also never negative. It is always equal to the magnitude of the actual vector. Here are some examples to understand this.

• 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
• We have seen that if v = <x, -y> then -v = <-x, y>. Here,
  |v| = √(x² + (-y)²) = √(x² + y²)
  |v| = √((-x)² + y²) = √(x² + y²)

In each of these examples, the magnitude of the vector is equal to the magnitude of its negative vector.

• In dot product, - a · b = a · (-b) = -1(a · b).
• In cross product, - a × b = a × (-b) = -1(a × b).
• Also, a × b = - (b × a). i.e., a × b and - (b × a) are negative vectors of each other.
• Further, the cross product of a vector and its negative vector is a zero vector. i.e., a × (-a) = - (a × a) = 0.
• The sum of a vector and its negative vector is a zero vector. i.e., a + (-a) = 0, for any vector a.

☛ Related Topics:

• Product of Vectors
• Unit Vector
• Vector Quantities
• Vector Formulas
• Skew Lines

How Do You Find the Negative of a Vector?

To find the negative of a vector, we multiply it by -1. i.e., literally, we are multiplying each of its components by -1 (or) in other words, we just need to change the sign of each of its components. For example, the negative of a vector p = <-5, 6> is -p = <5, -6>.

Which is True About the Negative of a Vector?

The negative of a vector is in the opposite direction of the given vector. The vector and its negative always have the same magnitudes.

Is a Negative Number a Vector?

No, a negative number is a scalar always. If a vector is multiplied by -1 (or) all its components' signs are changed, then the vector obtained is the negative vector of the given vector.

Does it Make Sense to Say that a Vector is Negative?

No, it doesn't make sense to say that "a vector is negative", rather it makes sense to say "a vector is a negative of another vector". For example, we cannot say that - a is negative, rather we say that -a is the negative of a vector a.

What is the Magnitude of a Negative Vector?

The magnitude of a negative of a vector is always equal to that of the original vector. If v and -v are negatives of each other then |v| = |-v|.