The negative of a vector is another vector that has the same magnitude but the opposite direction. If A is a vector, then its negative is written as –A.
To find the negative of a vector, simply reverse the direction of the original vector while keeping the magnitude unchanged. Algebraically, if A = (x, y, z), then –A = (–x, –y, –z).
When a vector is added to its negative, the result is the zero vector. For example, A + (–A) = 0.
No. The negative of a vector is a vector pointing in the opposite direction, while subtraction involves combining one vector with the negative of another. For example, A – B = A + (–B).
No. The negative of a vector only changes its direction, not its magnitude.
Geometrically, the negative of a vector is represented by an arrow of the same length as the original vector but pointing exactly opposite to it.
Join ALLEN!
(Session 2026 - 27)
Choose class
Choose your goal
Preferred Mode
Choose State
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, A=a1i^+a2j^+a3k^, then the negative of vector A is: −A=−a1i^−a2j^−a3k^
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 A=3i^+4j^,then−A=−3i^−4j^
Notice that : A+(−A)=0
3.0Geometrical Representation of Negative Vectors
The geometrical representation of the negative of a vector is straightforward:
Draw the original vector A from point O to P.
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., m−1.
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: 23
The slope of the perpendicular vector is −32, 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
Equal Magnitude: The negative of a vector has the same magnitude as the original vector.(|A| = |–A| )
Opposite Direction:Aand−A are collinear but point in opposite directions.
Zero Vector on Addition: Adding a vector and its negative results in the zero vector A+(−A)=0
Scalar Multiplication: If k is a scalar, then: k(−A)=−(kA)