Parallel (Collinear) Vectors

Statement

Two vectors are said to be parallel (or collinear) if they lie along the same line, even if pointing in opposite directions. Mathematically, vectors \(\vec{a}\) and \(\vec{b}\) are parallel if there exists a scalar \(k \in \mathbb{R}\) such that:

\[ \vec{a} = k \vec{b} \]

If \(k > 0\), the vectors point in the same direction. If \(k < 0\), the vectors point in opposite directions. If no such scalar exists, the vectors are not parallel.

Why it’s true

• A vector is defined by both magnitude and direction. Two vectors are parallel if their directions match (or are exactly opposite).
• Multiplying a vector by a scalar stretches or shrinks its length without changing its direction — hence, scalar multiples indicate parallel vectors.

Recipe (how to use it)

1. Write the vectors in component form, e.g., \(\vec{a}=(a_1, a_2)\), \(\vec{b}=(b_1, b_2)\).
2. Check if \(\frac{a_1}{b_1} = \frac{a_2}{b_2}\) (and for 3D, also \(\frac{a_3}{b_3}\)).
3. If the ratios are equal, the vectors are parallel; the common ratio is the scalar \(k\).
4. If the ratios differ, the vectors are not parallel.

Spotting it

You are often asked to check if two vectors are parallel, or to find a missing value so that they are parallel. This usually involves solving for \(k\) or checking equal ratios between components.

Common pairings

• Coordinate geometry: proving points lie on the same line.
• Forces in physics: forces in the same or opposite direction.
• Collinearity of three points (via vectors between them).

Mini examples

1. Given: \(\vec{a}=(2,4)\), \(\vec{b}=(1,2)\). Find: Are they parallel? Answer: Yes, \(\vec{a}=2\vec{b}\).
2. Given: \(\vec{a}=(3,-6)\), \(\vec{b}=(-1,2)\). Find: Are they parallel? Answer: Yes, \(\vec{a}=-3\vec{b}\).

Pitfalls

• Dividing by zero — if one component is zero, handle carefully.
• Assuming parallel when only one pair of components matches; all ratios must be equal.
• Forgetting that negative scalars also indicate parallel vectors, just opposite directions.

Exam strategy

• Write vectors in simplest form to check proportionality easily.
• Look out for “show that points are collinear” — this means the vectors between them are parallel.
• In 3D, check all three component ratios — missing one leads to errors.

Summary

Vectors are parallel if one is a scalar multiple of the other. The test is simple: all component ratios must be the same. Positive multiples point the same way, negative multiples point opposite ways. Recognising this property is key in vector geometry and mechanics problems.