Negative of a Vector

1.0Negative of a Vector Definition

The negative of a vector is defined as a vector having the same magnitude as the original vector but pointing in the exact opposite direction. If A is a vector, its negative is denoted by –A.

Mathematically:

If, $\vec{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, then the negative of vector $\vec{A}$ is: $-\vec{A}=-a_1\hat{i}-a_2\hat{j}-a_3\hat{k}$

This means each component of the vector is multiplied by –1.

2.0What Does Negative of a Vector Mean?

The negative of a vector means:

• The new vector is oriented in the opposite direction to the original vector.
• Its magnitude remains unchanged.
• When the original vector and its negative are added, the result is the zero vector.

For example: If $\vec{A}=3\hat{i}+4\hat{j},then-\vec{A}=-3\hat{i}-4\hat{j}$

Notice that : $\vec{A}+(-\vec{A})=0$

3.0Geometrical Representation of Negative Vectors

The geometrical representation of the negative of a vector is straightforward:

1. Draw the original vector A from point O to P.
2. The negative vector –A is drawn from the same origin O, but in the opposite direction to A and with the same length.

If A = OP, then –A = OQ such that OP = OQ in magnitude but OQ is directed oppositely to OP.

Visualization:

• If A points upward, –A will point downward.
• If A points north, –A points south.

This graphical approach helps understand how vectors interact, especially in physical problems involving forces or velocities.

4.0Negative Reciprocal of a Vector

The phrase negative reciprocal of a vector is most often used in the context of slopes and orthogonality.

• In two dimensions, if a vector has slope m, then the slope of a line perpendicular to it is the negative reciprocal, i.e., $\frac{-1}{m}$​.
• This is used in coordinate geometry and dot product properties, since two vectors are perpendicular if their dot product equals zero.

Example:
If a vector has direction ratio (2, 3), its slope is: $\frac{3}{2}$

The slope of the perpendicular vector is $-\frac{2}{3}$, which is the negative reciprocal.

So, the negative reciprocal of a vector helps us define perpendicularity in 2D vector problems.

5.0Properties of Negative of a Vectors

1. Equal Magnitude: The negative of a vector has the same magnitude as the original vector.(|A| = |–A| )
2. Opposite Direction: $\vec{A}and-\vec{A}$ are collinear but point in opposite directions. 
3. Zero Vector on Addition: Adding a vector and its negative results in the zero vector $\vec{A}+(-\vec{A})=\vec{0}$
4. Scalar Multiplication: If k is a scalar, then: $k(-\vec{A})=-(k\vec{A})$

Also Read: Vector Algebra

6.0Solved Examples on Negative of a Vectors

Example 1: Negative of a 2D Vector

Q. If $\vec{A}=4\hat{i}-5\hat{j}$, find $-\vec{A}$

Solution:

$-\vec{A}=-4\hat{i}+5\hat{j}$

Magnitude:

$|\vec{A}|=\sqrt{4^2+(-5{)}^2}=\sqrt{41}$

$|-\vec{A}|=\sqrt{(-4{)}^2+(5{)}^2}=\sqrt{41}$

Both have the same magnitude.

Example 2: Negative Reciprocal in Geometry

Find the slope of a line perpendicular to vector ​$\vec{A}=2\hat{i}+3\hat{j}$.

Solution:

Slope of $\vec{A}=\frac{3}{2}$.

Slope of perpendicular line = negative reciprocal $=-\frac{2}{3}$

Example 3: Opposite Vectors in 3D

If $\vec{B}=2\hat{i}+3\hat{j}+\hat{k}$, then $-\vec{B}=-2\hat{i}-3\hat{j}-\hat{k}$

Adding them:

$\vec{B}+(-\vec{B})=\vec{0}$

Also Explore: Vector Algebra Previous Year Questions with Solutions

7.0Practice Questions on Negative of a Vectors

1. If $\vec{P}=7\hat{i}-2\hat{j}$ write its negative vector.
2. Show that $\vec{Q}+(-\vec{Q})=0,for\vec{Q}=3\hat{i}+4\hat{j}$
3. Find the negative reciprocal slope for the vector $\vec{R}=5\hat{i}+12\hat{j}$
4. If $\vec{S}=\hat{i}-\hat{j}+2\hat{k}$ compute $-\vec{S}$ and verify its magnitude equals that of $\vec{S}$.
5. Explain with an example how negative vectors are used in physics to represent forces.