Q.Answer the following as true or false.

Concept: Collinear Vectors Properties — Vectors are collinear if they lie along the same or parallel lines, regardless of magnitude or direction.

(i) $\vec{a}$ and $-\vec{a}$ lie along the same line (opposite directions).

True.

(ii) Collinear vectors can have different magnitudes (e.g., $2\vec{a}$ and $\vec{a}$).

False.

(iii) Same magnitude does not imply same or parallel direction (e.g., two vectors at $90^\circ$). …

The key idea here is simple: collinear vectors are vectors that lie along the same line or parallel lines. That means their directions are either exactly the same or exactly opposite. Magnitude has nothing to do with collinearity — two vectors can be collinear even if one is twice as long as the other.

Let’s go through each statement one by one.

(i) $\vec{a}$ and $-\vec{a}$ are collinear.

$\vec{a}$ and $-\vec{a}$ point in exactly opposite directions. But opposite directions still lie on the same straight line — one is just the reverse of the other. So they are collinear.

Result: True.

(ii) Two collinear vectors are always equal in magnitude.

Collinear vectors can have any length. For example, $\vec{a} = 3\hat{i}$ and $\vec{b} = 5\hat{i}$ are collinear (both along the x-axis), but their magnitudes are $3$ and $5$ — not equal. So this statement is false.

Result: False.

(iii) Two vectors having same magnitude are collinear. …

Method: Judging true/false statements about collinear vectors

Use this for conceptual true/false items testing collinearity, magnitude and equality.

Steps

Step 1: Anchor on the definition

Vectors are collinear when they lie along the same or parallel lines — i.e. one is a scalar multiple of the other, $\vec{a}=\lambda\vec{b}$. Direction sign and magnitude are free.

Step 2: Separate the three independent ideas

• Collinearity = same line of action (any sign, any length).
• Equal magnitude = same length (says nothing about direction).
• Equality = same magnitude and same direction.

Step 3: Test each statement with a quick counterexample …

Common Mistakes

Mistake 1: Believing collinear vectors must have equal magnitude

Why it's wrong: $\vec a=\lambda\vec b$ allows any scaling, so $2\hat i$ and $5\hat i$ are collinear with different lengths. Correct approach: collinearity constrains direction (same line), never magnitude.

Mistake 2: Thinking equal magnitude implies collinear

Why it's wrong: $5\hat i$ and $5\hat j$ share length $5$ but are perpendicular. Correct approach: magnitude alone says nothing about direction. …