Q.In the figure (Fig 10.6), four vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$ are shown. Identify the following vectors:

Recall: coinitial $=$ same starting point; equal $=$ same magnitude and direction; collinear $=$ along the same (or a parallel) line.

Reading Fig 10.6:

1. Coinitial: $\vec{a}$ and $\vec{d}$ start from the same point.
2. Equal: $\vec{b}$ and $\vec{d}$ have the same magnitude and the same direction, so $\vec{b}=\vec{d}$. …

The three tests

• Coinitial vectors share the same initial point (tail), regardless of length or direction.
• Equal vectors have both the same magnitude and the same direction (they may start anywhere).
• Collinear vectors lie along the same line or parallel lines; they may point the same way or opposite ways and may have different lengths.

Reading the figure

1. Coinitial — looking at the tails, $\vec{a}$ and $\vec{d}$ begin at the same point, so they are coinitial.
2. Equal — $\vec{b}$ and $\vec{d}$ are drawn with the same length and pointing the same way, so $\vec{b}=\vec{d}$. …

Method: Reading a figure to classify types of vectors

Use this whenever a diagram asks you to pick out coinitial, equal, or collinear vectors.

Steps

Step 1: Fix the three definitions before looking

• Coinitial — share the same initial point (tail); lengths/directions may differ.
• Equal — same magnitude and same direction (starting points irrelevant).
• Collinear — lie along the same or parallel lines; may differ in length or point oppositely.

Step 2: For coinitial, inspect only the tails

Find the arrows whose tails meet at a common point — ignore where their tips land.

Step 3: For equal, compare both length and sense …

Common Mistakes

Mistake 1: Equating "collinear" with "equal"

Why it's wrong: collinear only means same line of action; equal additionally demands identical length and sense. Arrows can be collinear yet point opposite ways or have different lengths. Correct approach: for equality, check both magnitude and direction.

Mistake 2: Thinking coinitial vectors must be equal (or vice versa) …