Two vectors are collinear (also called parallel) when they lie along the same straight line or along parallel lines — that is, they point in the same direction or in exactly opposite directions. Their lengths need not match; only their line of action must be the same.
Note
Because a vector can be slid freely without changing it, "same line" and "parallel lines" mean the same thing for collinearity — direction is what counts.
The key property: one is a scalar multiple of the other
The defining test is beautifully simple. Two vectors a and b (with b=0) are collinear if and only if there is a scalar λ such that
a=λb
If λ>0, they point the same way.
If λ<0, they point in opposite ways.
∣λ∣ tells you how many times longer a is than b.
In component form
If a=a1i^+a2j^+a3k^ and b=b1i^+b2j^+b3k^, then a=λb forces each component to match, so their components are proportional:
b1a1=b2a2=b3a3=λ.
Other useful properties
The zero vector is collinear with every vector (take λ=0).
Collinearity can also be tested with the cross product: a and b are collinear ⟺a×b=0, since parallel vectors enclose a zero-area parallelogram.
Three points A,B,C are collinear ⟺AB and AC are collinear vectors. …
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2026Set 65/3/11 markMCQ
Q.Assertion (A): Vectors a and (−2a), where a=0, are collinear vectors. Reason (R): a⋅(−2a)=0.
›Reveal solutionSolution
The assertion is true because one vector is a scalar multiple of the other, making them collinear. The reason is false because the dot product of a and −2a is −2∣a∣2, not zero. So (A) is true, (R) is false.
Let’s unpack this carefully. The question tests two separate ideas: what it means for vectors to be collinear, and what the dot product of a vector with its scalar multiple actually gives. Many students mix these up, so let’s build the intuition first.
Collinear vectors are vectors that lie along the same line or parallel lines. The key property: two non-zero vectors are collinear if and only if one is a scalar multiple of the other. That is, b=λa for some real number λ. No angle condition or dot product zero is required — that’s for perpendicular vectors.
Now look at the assertion: a and (−2a) are clearly related by λ=−2. Since −2 is a real number, they are indeed collinear. The negative sign just means they point in opposite directions, but they’re still on the same line.
The reason claims a⋅(−2a)=0. Let’s compute that:
a⋅(−2a)=−2(a⋅a)=−2∣a∣2.
Since a=0, ∣a∣2>0, so this product is negative, not zero. The reason is completely wrong.
Watch out
A common mistake is to think that collinear vectors have a dot product of zero. That’s actually the condition for perpendicular (orthogonal) vectors. Collinearity is about scalar multiples, not dot products.
So the assertion is true, the reason is false. In exam language, this means option (C) — Assertion true, Reason false.
Tip
Quick check: If two vectors are collinear, their dot product equals ±∣a∣∣b∣, not zero. Zero dot product only happens when they are perpendicular (or one is zero).
Collinear vectors are vectors parallel to one and the same line.
Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions. Equivalently, a and b are collinear if a=λb for …