Opposite Vector

Reversing a Vector without Changing Its Length

For a nonzero vector $\vec{v}$ , its opposite vector has the same magnitude but points in exactly the opposite direction. It is written as $-\vec{v}$ .

If $\vec{v}$ describes a displacement from left to right, then $-\vec{v}$ describes an equally long displacement from right to left. Their lengths match, and their directions are opposite.

Formula for an Opposite Vector

Multiplying a vector by the scalar $-1$ reverses every component:

-\vec{v}=(-1)\vec{v}

If $\vec{v}=(v_1,v_2,v_3)$ , then:

-\vec{v}=(-v_1,-v_2,-v_3)

Here:

$\vec{v}$ is the original vector.
$-\vec{v}$ is its opposite vector.
The factor $-1$ reverses the direction without changing the magnitude.

Finding the Opposite of a Vector

Suppose $\overrightarrow{AB}=6\mathbf{i}+8\mathbf{j}$ .

Its magnitude is:

|\overrightarrow{AB}|=\sqrt{6^2+8^2}=\sqrt{100}=10

Multiply every component by $-1$ to find the opposite vector:

-\overrightarrow{AB}=-(6\mathbf{i}+8\mathbf{j})=-6\mathbf{i}-8\mathbf{j}

Both vectors have magnitude $10$ , but their directions differ by $180^\circ$ .

A Vector and Its Opposite

The vectors

\overrightarrow{AB}

and

-\overrightarrow{AB}

have equal lengths and point in opposite directions.

Properties of Opposite Vectors

Their magnitudes are equal:

$|-\vec{v}|=|\vec{v}|$
For a nonzero vector, their directions differ by $180^\circ$ .
Their sum is the zero vector:

$\vec{v}+(-\vec{v})=\vec{0}$
Taking the opposite twice returns the original vector:

$-(-\vec{v})=\vec{v}$

The zero vector is its own opposite because $-\vec{0}=\vec{0}$ .

Applications of Opposite Vectors

Opposite vectors appear whenever a direction is reversed:

Displacement: a return trip along the same straight path has the opposite displacement vector.
Force equilibrium: two equal forces in opposite directions add to the zero vector.
Vector equations: subtraction can be rewritten as addition of an opposite vector, $\vec{a}-\vec{b}=\vec{a}+(-\vec{b})$ .

Unit Vectors and Opposite Direction

A unit vector is built from a nonzero vector by dividing that vector by its length. Multiplying the result by $-1$ reverses its direction while keeping the same line.

Unit Vector

For a nonzero vector $\vec{v}$ , its unit vector is:

\hat{v}=\frac{\vec{v}}{|\vec{v}|}

It has magnitude $1$ and points in the same direction as $\vec{v}$ .

Opposite Unit Directions

The two unit vectors have equal length and opposite direction.

Relationship with the Opposite Vector

The unit vector in the opposite direction is $-\hat{v}$ . Multiplying it by the original magnitude gives the opposite vector:

-\vec{v}=|\vec{v}|(-\hat{v})

This separates the two ideas clearly: $|\vec{v}|$ determines the length, while $-\hat{v}$ determines the opposite direction.