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Opposite Vector

In vector algebra, an opposite vector is characterized by having the same direction and magnitude as the original, but extends in the opposite direction.

Consider the following example for clarity.

Take a vector u that links the plane's origin to the point at the coordinates (3,1). This is a typical vector.

example of vector

 

The antithesis of this vector u is another vector that maintains the same direction (the same angle of inclination) and identical magnitude (length), but points in the opposite direction.

So, the opposite vector of u will connect the plane's origin to a point, but this time located at the coordinates (-3,-1).

the opposite vector of u

Terminology-wise, the opposite vector of u is represented as -u. It shares the same coordinates as u, however, the algebraic signs are flipped, thus becoming (-x,-y).

The sum of two opposite vectors always equals the zero vector.

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

For instance, the vector \( \vec{u} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \) has an opposite vector \( -\vec{u} = \begin{pmatrix} -3 \\ -2 \end{pmatrix} \).

When you add the two vectors together, their components cancel out, resulting in a zero vector:

$$ \vec{u} + (-\vec{u}) = \begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} -3 \\ -2 \end{pmatrix} = \begin{pmatrix} 3-3 \\ 2-2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$

We trust this lesson from Nigiara proves insightful. We invite you to continue to engage with our material.




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