from a handpicked tutor in LIVE 1to1 classes
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.
1.  What is the Negative of a Vector? 
2.  How to Find Negative of a Vector? 
3.  Magnitude of Negative of a Vector 
4.  Properties of Negative of a Vector 
5.  FAQs on Negative of a Vector 
What is the Negative of a Vector?
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.
How to Find Negative of a Vector?
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.
Magnitude of Negative of a Vector
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 = √(1^{2} + (2)^{2} + (3)^{2}) = √(1+4+9) = √14
a = √((1)^{2} + 2^{2} + 3^{2}) = √(1+4+9) = √14  We have seen that if v = <x, y> then v = <x, y>. Here,
v = √(x^{2} + (y)^{2}) = √(x^{2} + y^{2})
v = √((x)^{2} + y^{2}) = √(x^{2} + y^{2})
In each of these examples, the magnitude of the vector is equal to the magnitude of its negative vector.
Properties of Negative of a 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:
Negative of a Vector Examples

Example 1: If a vector is p = <3, 7, 1>, find the negative of p.
Solution:
To find the negative of a vector, just change the signs of its components. Then
p = <3, 7, 1>
Answer: The negative of p is <3, 7, 1>.

Example 2: If <x, y> is the negative of a vector <7, 10>, then what is x + y?
Solution:
The negative of <7, 10> is <7, 10>.
So <7, 10> = <x, y>
From this, x = 7 and y = 10 (or) y = 10.
So x + y = 7 + (10) = 3.
Answer: x + y = 3.

Example 3: Determine the pairs of negative vectors in the following parallelogram.
Solution:
A vector and its negative always have same magnitude but they are in opposite directions.
Thus, the pairs of negative vectors in the above parallelogram are PQ and RS; and SP and QR.
Answer: PQ and RS; and SP and QR.
FAQs on Negative of a Vector
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.
visual curriculum