Chapter 10 · Vector Algebra

Example 3: Collinear, equal and coinitial vectors in Fig 10.5

ExamplesTypes of vectors · Section 10.3

The question

In Fig 10.5, which of the vectors are:

(i) Collinear (ii) Equal (iii) Coinitial

Exam answer

In the figure, let Q be the point where $\vec b$ ⁠, $\vec c$ and $\vec d$ start, and P the point where $\vec b$ ends and $\vec a$ starts.

(i) $\vec a$ ⁠, $\vec c$ and $\vec d$ are parallel to the same line ( $\vec b$ is not), so $\vec a$ ⁠, $\vec c$ and $\vec d$ are collinear.

(ii) $\vec a$ and $\vec c$ have the same magnitude (2 units) and the same direction, so $\vec a = \vec c$ . ( $\vec d$ has the opposite direction.)

(iii) $\vec b$ ⁠, $\vec c$ and $\vec d$ have the same initial point Q, so $\vec b$ ⁠, $\vec c$ and $\vec d$ are coinitial.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

Three definitions from Section 10.3, each applied by looking at the arrows. Collinear: parallel to the same line, whatever their magnitudes and directions. Equal: same magnitude and same direction, wherever they start. Coinitial: the same initial point. So for each arrow we read off three things: where it starts, which way it points and how long it is (from the tick marks, 1 unit each).

Given, to find, and the plan

Given: Fig 10.5: four vectors $\vec a, \vec b, \vec c, \vec d$ with tick marks of 1 unit (redrawn below). Three of them, $\vec b, \vec c, \vec d$ ⁠, start at one point; $\vec a$ starts where $\vec b$ ends.
To find: (i) the collinear vectors, (ii) the equal vectors, (iii) the coinitial vectors.
Plan: (1) Name the two marked points and describe each arrow. (2) Group the arrows by the line they are parallel to, for (i). (3) Among parallel arrows, compare direction and length, for (ii). (4) Compare initial points, for (iii). (5) Confirm with components.

Basics you need

1Initial and terminal points

For $\overrightarrow{AB}$ ⁠, A (where the arrow starts) is the initial point and B (where the arrowhead is) is the terminal point (Section 10.2). Lesson: a vector is a length with a direction

2Collinear, equal and coinitial vectors

Collinear: parallel to the same line, irrespective of their magnitudes and directions. Equal: the same magnitude and direction, regardless of the positions of their initial points. Coinitial: the same initial point (Section 10.3). Lesson: six kinds of vectors worth naming

3Free vectors

A vector may be moved parallel to itself without changing its magnitude and direction (Remark, Section 10.3). That is why two arrows with different starting points can still be equal.

4Collinear means a scalar multiple

$\vec b$ is collinear with a nonzero $\vec a$ exactly when $\vec b = \lambda\vec a$ for some scalar $\lambda$ (Remark (i), Section 10.5.1). Lesson: components

Part (i): collinear vectors

Name the points and read each arrow

Call the point where three arrows start Q, and the other marked point P. Reading the figure:

Fig 10.5 redrawn to scale (1 unit = one tick length). From Q: $\vec b$ runs to P (3 units), $\vec c$ runs up the line through Q (2 units) and $\vec d$ runs down the same line (3 units). From P: $\vec a$ runs up a line parallel to that line (2 units).

$\vec b$ ⁠: from Q to P, 3 units. $\vec c$ ⁠: from Q, up and slightly to the left, 2 units. $\vec d$ ⁠: from Q, down and slightly to the right, along the same line as $\vec c$ ⁠, 3 units. $\vec a$ ⁠: from P, up and slightly to the left, parallel to $\vec c$ ⁠, 2 units.

Every question in this example is about where arrows start, which way they point and how long they are, so we collect those three facts first. The tick marks split $\vec a$ and $\vec c$ into 2 equal parts and $\vec b$ and $\vec d$ into 3, at the figure's scale of 1 unit.

The arrowhead marks the terminal point; the other end is the initial point (Section 10.2). Lesson: a vector is a length with a direction
Group the arrows by their line

$\vec c$ and $\vec d$ lie on one line through Q, and $\vec a$ lies on a line through P parallel to it. $\vec b$ runs across, at right angles to that direction, so it is parallel to none of the others.

So $\vec a$ ⁠, $\vec c$ and $\vec d$ are collinear.

Collinear vectors only need to be parallel to the same line; their magnitudes and directions do not matter. So $\vec d$ ⁠, which points the other way and is longer, still belongs with $\vec a$ and $\vec c$ ⁠. And $\vec a$ need not lie on the same actual line as $\vec c$ ⁠: being parallel to it is enough.

"Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions" (Section 10.3). Lesson: six kinds of vectors worth naming

Part (ii): equal vectors

Compare direction and length

Equal vectors must point the same way, so they must first be collinear; we look only at $\vec a, \vec c, \vec d$ ⁠. $\vec a$ and $\vec c$ both point up the line, and both are 2 units long. $\vec d$ points down the line (and is 3 units long).

So $\vec a = \vec c$ , and no other pair is equal.

$\vec a$ and $\vec c$ start at different points (P and Q), but equality is "regardless of the positions of their initial points": slide $\vec a$ parallel to itself from P to Q and it lies exactly on $\vec c$ ⁠. $\vec d$ fails on direction (and length), and $\vec b$ fails on direction, so neither equals anything else.

Equal vectors: the same magnitude and direction (Section 10.3); vectors in this chapter are free vectors (Remark, Section 10.3). Lesson: six kinds of vectors worth naming

Part (iii): coinitial vectors

Compare initial points

$\vec b$ ⁠, $\vec c$ and $\vec d$ all start at Q. $\vec a$ starts at P, which is where $\vec b$ ends.

So $\vec b$ ⁠, $\vec c$ and $\vec d$ are coinitial.

Coinitial depends only on the starting point, not on direction or length: $\vec b$ ⁠, $\vec c$ and $\vec d$ point three different ways but share the initial point Q. $\vec a$ is left out because its initial point is P; P is the terminal point of $\vec b$ ⁠, not its initial point.

"Two or more vectors having the same initial point are called coinitial vectors" (Section 10.3). Lesson: six kinds of vectors worth naming
Confirm with components

Take Q as origin and unit vectors along the two directions in the figure: $\vec e = \frac{1}{\sqrt{10}}(-\hat i + 3\hat j)$ up the line of $\vec c$ ⁠, and $\vec f = \frac{1}{\sqrt{10}}(-3\hat i - \hat j)$ from Q towards P. Then
$\vec a = 2\vec e,\quad \vec c = 2\vec e,\quad \vec d = -3\vec e,\quad \vec b = 3\vec f.$
So $\vec c - \vec a = \vec 0$ (equal), $\vec d = -\frac32\,\vec a$ (collinear, opposite direction), and $\vec b$ is not a multiple of $\vec a$ ⁠: $\vec b = \lambda\vec a$ would need $-9 = -2\lambda$ and $-3 = 6\lambda$ from the $\hat i$ and $\hat j$ parts (after multiplying by $\sqrt{10}$ ⁠), that is $\lambda = \frac92$ and $\lambda = -\frac12$ at once, which is impossible.

Components turn the three visual checks into arithmetic. The two directions are perpendicular: $(-1)(-3) + (3)(-1) = 0$ ⁠. They were measured from the book's figure, where they agree with the printed arrows to within half a degree, so the model is a faithful copy of Fig 10.5.

Two vectors are equal exactly when their components are equal, and collinear exactly when one is a scalar multiple of the other (Section 10.5.1 and its Remark (i)). Lesson: components

Result

(i) Collinear: $\vec a$ ⁠, $\vec c$ and $\vec d$ ⁠. (ii) Equal: $\vec a$ and $\vec c$ ⁠. (iii) Coinitial: $\vec b$ ⁠, $\vec c$ and $\vec d$ ⁠.

(Textbook answer: (i) Collinear vectors: $\vec a$ ⁠, $\vec c$ and $\vec d$ ⁠. (ii) Equal vectors: $\vec a$ and $\vec c$ ⁠. (iii) Coinitial vectors: $\vec b$ ⁠, $\vec c$ and $\vec d$ ⁠.)

Check it with numbers

In the component model: $|\vec a| = |\vec c| = 2$ ⁠, $|\vec b| = |\vec d| = 3$ ⁠. In decimals, $\vec a = \vec c \approx -0.6325\hat i + 1.8974\hat j$ and $\vec d \approx 0.9487\hat i - 2.8460\hat j$ ⁠; the ratio of the $\hat i$ parts $0.9487 \div (-0.6325) \approx -1.5$ equals the ratio of the $\hat j$ parts $-2.8460 \div 1.8974 \approx -1.5$ ⁠, as collinearity requires. For $\vec b \approx -2.8460\hat i - 0.9487\hat j$ the ratios are $-2.8460 \div (-0.6325) \approx 4.5$ and $-0.9487 \div 1.8974 = -0.5$ ⁠: different, so $\vec b$ is not collinear with $\vec a$ ⁠.

The directions in the book's figure: $\vec a$ and $\vec c$ each make about $108.2^\circ$ with the positive x-direction, $\vec d$ about $-71.8^\circ$ (the opposite direction, $108.2^\circ - 180^\circ$ ⁠), and $\vec b$ about $198.3^\circ$ ⁠; the model's $\vec e$ makes $108.43^\circ$ and $\vec f$ makes $198.43^\circ$ ⁠.

Common mistakes

Leaving $\vec d$ out of the collinear set because it points the other way. Collinear vectors may have any directions, as long as they are parallel to one line.
Leaving $\vec a$ out of the collinear set because it is on a different line. Parallel lines are enough; the vectors are free.
Calling $\vec c$ and $\vec d$ equal because they are on the same line. Equal needs the same direction and the same magnitude.
Calling $\vec a$ coinitial with $\vec b$ because they touch at P. P is the end of $\vec b$ ⁠, not its start.

What to take away

For each arrow, read three things: start, direction, length.
Collinear: parallel to one line (any direction, any length). Equal: same direction and same length (any start). Coinitial: same start (any direction, any length).
Equal vectors are always collinear; collinear vectors need not be equal.

Every fact used, and where it comes from

Fact	Used in	Source
Initial point and terminal point of a vector	Part (i), Step 1; Part (iii), Step 1	Section 10.2
Collinear, equal and coinitial vectors	Part (i), Step 2; Part (ii), Step 1; Part (iii), Step 1	Section 10.3 (Lesson: six kinds of vectors)
Free vectors may be moved parallel to themselves	Part (ii), Step 1	Remark, Section 10.3
Equality by components; $\vec b$ collinear with $\vec a$ iff $\vec b = \lambda\vec a$	Part (iii), Step 2	Section 10.5.1 and its Remark (i) (Lesson: components)
The figure and the book's answer	Part (i), Step 1; Result	Example 3 and Fig 10.5, Section 10.3 (Lesson: Examples 1–3)

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Example 3: Collinear, equal and coinitial vectors in Fig 10.5

Basics you need

Part (i): collinear vectors

Name the points and read each arrow

Group the arrows by their line

Part (ii): equal vectors

Compare direction and length

Part (iii): coinitial vectors

Compare initial points

Confirm with components

Every fact used, and where it comes from