Q.In Fig 10.5, which of the vectors are:

Read the four vectors from the figure. $\vec{a}$ points up-right with length $2$. From the common point $M$: $\vec{c}$ points up-right (length $2$), $\vec{d}$ points down-left (length $1$) along the same line, and $\vec{b}$ points down-right (length $1$) in a different direction.

• (i) Collinear (parallel to one line): $\vec{a}$, $\vec{c}$ and $\vec{d}$ all lie along the same line, so $\vec{a},\ \vec{c},\ \vec{d}$ are collinear. …

The three ideas

• Collinear vectors are parallel to the same line — their directions are the same or exactly opposite. Length and starting point do not matter.
• Equal vectors have the same magnitude and the same direction (position is irrelevant).
• Coinitial vectors have the same initial (starting) point.

What the figure shows

• $\vec{a}$: a free vector pointing up-and-to-the-right, length $2$.
• $\vec{c}$: starts at $M$, points up-and-to-the-right, length $2$ — same line and direction as $\vec{a}$.
• $\vec{d}$: starts at $M$, points down-and-to-the-left, length $1$ — along the same line as $\vec{a}$ and $\vec{c}$ but opposite in direction.
• $\vec{b}$: starts at $M$, points down-and-to-the-right, length $1$ — a different direction, not along that line.

Apply each definition

1. Collinear. $\vec{a}$, $\vec{c}$ and $\vec{d}$ are all parallel to one line ($\vec{a},\vec{c}$ in one direction, $\vec{d}$ opposite). $\vec{b}$ points a different way, so it is excluded. Collinear: $\vec{a},\ \vec{c},\ \vec{d}$. …

Method: Classifying vectors as collinear, equal, or coinitial from a figure

Use this whenever a diagram of several arrows asks you to sort them into these three (independent) categories.

Steps

Step 1: Fix the precise definition of each label.

• Collinear — parallel to one common line (same or exactly opposite direction); length and starting point do not matter. Test: $\vec{a}=\lambda\vec{b}$ for some scalar $\lambda$.
• Equal — same magnitude and same direction; position is irrelevant.
• Coinitial — sharing the same initial point (tail); direction and length are irrelevant.

Step 2: Test every arrow against each definition separately. …

Common Mistakes

Mistake 1: Assuming vectors from the same point are automatically equal.

Why it's wrong: coinitial means same tail only — $\vec{b},\vec{c},\vec{d}$ share point $M$ but differ in direction/length, so they are not equal. Correct approach: for equality demand identical magnitude AND direction, independent of starting point.

Mistake 2: Calling collinear vectors equal. …