Antiparallel — Definition, Formula & Examples

Antiparallel describes two lines or vectors that lie along the same direction (or parallel directions) but point in opposite senses. In triangle geometry, two lines are antiparallel with respect to an angle if they make equal angles with the two sides of that angle, but in the reverse order compared to parallel lines.

Two directed lines (or vectors) are antiparallel if they are parallel and oppositely oriented. More generally, in the context of a triangle with vertex $A$ , a line segment is antiparallel to side $BC$ with respect to $\angle A$ if it makes the same angles with sides $AB$ and $AC$ as $BC$ does, but with the angles swapped — that is, if the segment and $BC$ form a cyclic quadrilateral with vertices on $AB$ and $AC$ .

How It Works

In the vector sense, if

\vec{u}

points in a given direction, then

-\vec{u}

is antiparallel to it. Any negative scalar multiple

-k\vec{u}

(with

k > 0

) is also antiparallel to

\vec{u}

. In triangle geometry, draw triangle

ABC

and consider any line crossing sides

AB

and

AC

. If this line makes

\angle ABD' = \angle ACB

and

\angle ACD' = \angle ABC

(the angles are swapped compared to

BC

), then the line is antiparallel to

BC

. A key property is that four points forming an antiparallel configuration are concyclic — they lie on a common circle.

Worked Example

Problem: Vectors

\vec{u} = \langle 3, 4 \rangle

and

\vec{v} = \langle -6, -8 \rangle

are given. Determine whether they are parallel, antiparallel, or neither.

Step 1: Check if one vector is a scalar multiple of the other.

\vec{v} = \langle -6, -8 \rangle = -2 \cdot \langle 3, 4 \rangle = -2\vec{u}

Step 2: Since the scalar is

-2

, which is negative, the vectors point in opposite directions. They are parallel in the undirected sense but oppositely oriented.

Answer:

\vec{v}

is antiparallel to

\vec{u}

because

\vec{v} = -2\vec{u}

, meaning they share the same line of action but point in opposite directions.

Why It Matters

Antiparallel lines appear in triangle geometry when studying the symmedian point, where symmedians are reflections of medians that create antiparallel segments. In physics, antiparallel vectors describe opposing forces or magnetic moments, such as electron spin configurations in antiferromagnetic materials.

Common Mistakes

Mistake: Treating antiparallel as simply meaning 'not parallel.'

Correction: Antiparallel specifically means parallel but in the opposite direction (for vectors) or making swapped equal angles with two reference lines (in triangle geometry). Two lines that merely intersect at a random angle are neither parallel nor antiparallel.