Negative of a Vector: Definition, Formula and Solved Examples

Negative of a vector is a type of vector that is in an opposite direction of a given vector. A vector, like a scalar, can be both positive and negative. Let us assume two Vectors, both vectors will be considered as the negative of each other if their magnitudes are the same but their directions are absolutely opposite. The direction opposite to the reference direction is represented by the negative of a vector. It signifies that two vectors have the same magnitude but opposing directions. In this article, we will discuss what a negative vector is and how to find the negative of a vector. 

Also Read: NCERT Solutions For Class 12 Mathematics Chapter 10 Vector Algebra 

Key Terms: Vectors, Magnitude, Negative Vector, Multiplications, Relation, Direction, Reference vector

Also Read: Vector product of two vectors

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What is the Negative of a Vector?

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Negative vectors are those vectors that have the same length as a given vector but point in the opposite direction. A negative sign turns a vector into a negative vector by reversing its direction. Only in relation to another vector can a vector be negative. Suppose there are two vectors a and b, which are precisely the same in magnitude but opposite in direction. These vectors can be represented as-

a = -a

Negative of a Vector

If a vector AB points left to right, for example, the vector BA will point right to left. Because the directions are diametrically opposed, we say AB = – BA. As shown in the diagram below, BA is the negative vector for AB. It's worth noting that the vectors AB and BA have the same magnitude but are oriented in opposite directions, making them negative vectors.

A vector's magnitude, or length, cannot be negative; it must be either zero or positive. The negative sign in front of a vector denotes that the vector is pointing in an opposite direction as compared to the reference vector.

Also Read: Addition of Vectors

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How to find the Negative of a Vector?

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Finding the negative vector of a given vector is as easy as locating the two components of the provided vector (i.e., magnitude and direction) and then locating a vector of the same length pointing in the opposite direction. Two such vectors will have the same negative vectors.

By putting a negative sign in front of a given vector, you can find its negative vector. Consider the case where P is a vector. We multiply P by -1 to get –P, which is the negative vector. Keep in mind that the magnitude of vector –P is equal to the magnitude of vector P.

The negative of a vector is obtained by reversing the direction of the supplied vector or multiplying the given vector by one.

Assume that a is the provided vector and that the negative of vector a is - a.

The magnitudes of a and -a will be the same but in different directions.

Also Read: Types of vectors

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Solved Examples

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Example 1 Determine which of the following vectors are equivalent and which are the inverses of each other: a = (1, 3), b = (-1, -3), and c = (1, 3).

Solution: We'll compare the magnitudes and directions of the given vectors to see which are equal and which are the inverses of one another.

The magnitude of the given vectors is first determined:

For the vector a = (1, 3)

|a| = √(1² + 3²)

|a| = √10

For the vector b = (-1, -3)

|b| = √(-1² + (-3)²)

|b| = √10

For the vector c = (1, 3)

|c| = √(1² + 3²) = √10

We can plot the three vectors on the coordinate plane to compare their directions, as illustrated in the image below. 

Three vectors on the coordinate plane

The vectors a and c have the same magnitude and also point in the same direction, as can be shown. In contrast, vector b points in the opposite direction. Hence, |a| = |b|, |a| = |c|, and |b| = |c| 

So, the magnitudes of the vectors a, b, and c are the same.

The vectors a and c are the same. i.e., a = c

The vector pairs a and b, as well as b and c, are both negative vectors.

So, a = -b and c = -b

Also Read: Multiplication of a vector by a scalar

Example 2 Determine the negative of A. Given the vector A = (2, 4). 

Solution: The magnitude of the negative of a vector is equal to the opposite direction of the reference vector by definition. The reference vector in this example is A, and its direction is 2 points right on the x-axis and 4 points up on the y-axis. To get A's negative vector, we multiply the reference vector A by -1 while keeping the magnitude constant. This gives us the following:

–A = (-2,-4) 

The negative vector's direction is two units to the left on the x-axis and 4 units down on the y-axis. This is plainly the polar opposite of reference vector A.

Example 3 Find the value of x for which the two vectors A = (4, 10) and B = (2x, 5x) are the inverses of each other.

Solution: When the magnitudes of two vectors are the same and their directions are opposite, we know they are the negatives of each other. This is how we figure out what the value of the unknown x is:

A = - B

⇒ (2, 10) = (2x, 5x)

We can reach the following result by putting the appropriate components equal to each other:

10 = - 5x, 2 = - 2x

When we simplify the equation above, we get x = - 2

As a result, when x = -2, the two vectors A and B are inverses of one another.

Also Read:

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Things to Remember

• A negative vector has the same magnitude as the reference vector but faces the opposite direction.
• When two vectors have the same magnitude but opposite directions, they are said to be negative.
• The magnitude, or length, of a vector, cannot be negative; it must be zero or positive.
• In the negative of a vector, the vector is pointing in the opposite direction as the reference vector, as shown by the negative sign.
• A vector with a magnitude higher than one can never be negative.

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Previous Year Questions

1. If \(\overrightarrow{a} and \overrightarrow{b}\) are unit vectors and… [KCET 2008]
2. f a and b are vectors such that… [KCET 2007]
3. If \(\overrightarrow{a}= 2\hat{i}+3\hat{j}-\hat{k}, \overrightarrow{b}= \hat{i}+2\hat{j}-5\hat{k}, \overrightarrow{c}= 3\hat{i}+5\hat{j}-\hat{k} \)… [KCET 2007] 
4. OA and BO are two vectors of magnitudes 5 and 6 respectively… [KCET 2007]
5. If a,b and c are three non-coplanar vectors and p,q and r are vectors defined by... [KCET 2012]
6. If a|(b+c) and a|(b−c) where a,b,c ∈ N then [KCET 2006]
7. The vector equation of the symmetrical form of equation of straight line… [VITEEE 2018]
8. If a is a unit vector, then… [TS EAMCET 2017]
9. Let \(\overrightarrow{a}, \overrightarrow{b}, \& \overrightarrow{c}\) be non-coplanar unit vectors equally inclined to one… [BITSAT 2017]
10. From a point A with position vector… [JEE Main 2018]
11. If a is a unit vector, then….​[TS EAMCET 2017]
12. Let a,b & c be non-coplanar unit vectors equally inclined to one another at an acute angle \thetaθ. Then |∣[abc]∣ in terms of θ is equal to….​.[BITSAT 2017]
13. Find the third vector C, if A+3B−C=0….[JKCET 2007]
14. The relationship between the angular velocity ω of the disc and number of rotations (n) made by the disc is governed by…
15. From a point A with position vector​...[JEE MAIN 2018]
16. Then its angular momentum, about the origin is perpendicular to...[KEAM]
17. If the volume of a parallelepiped whose coterminous edges are….[JEE MAIN 2020]