Lesson 2 of 6Vectors

Unit 5 · Lesson 2 of 6

Adding, Subtracting and Scaling Vectors

Add vectors tip to tail or as a parallelogram, subtract by adding the opposite, stretch with scalars, find magnitudes with the cosine law, and simplify vector expressions using the same properties as ordinary algebra.

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Key things to remember

1Triangle law: place the tail of $b$ on the head of $a$ ; $a + b$ runs from the first tail to the last head.
2 $a - b = a + (- b)$ ; tail to tail, it runs from the head of $b$ to the head of $a$ .
3 $∣ k a ∣ = ∣ k ∣∣ a ∣$ ; $k < 0$ reverses the direction. Non-zero $a$ and $b$ are collinear when $b = k a$ .
4With $θ$ the angle between them (tail to tail): $∣ a \pm b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} \pm 2∣ a ∣∣ b ∣ cos θ .$
5Addition is commutative and associative; scalar multiplication distributes over both vector and scalar addition.

Walk $3$ km east and then $4$ km north. You have walked $7$ km, but you are only $5$ km from where you started. Combining vectors is not adding their lengths. It is following one arrow and then the next and asking where you ended up. That single idea, done carefully, gives every rule in this lesson.

Adding: the triangle law

To find $a + b$ , draw $a$ , then slide $b$ (without turning it) so that its TAIL sits on the HEAD of $a$ . The sum, called the RESULTANT, is the arrow from the tail of $a$ to the head of $b$ . Sliding is allowed because equal vectors can sit anywhere. For more than two vectors, keep chaining tip to tail; the resultant joins the very first tail to the very last head.

Fig 1A 45 m north vector and a 60 m east vector drawn tip to tail. The 75 m resultant, about N 53 degrees E, closes the right triangle, while the distance walked is 105 m.

The PARALLELOGRAM LAW gives the same answer from a different picture: draw $a$ and $b$ tail to tail, complete the parallelogram, and the diagonal from the shared tail is $a + b$ . The two triangles inside the parallelogram show $a$ then $b$ and $b$ then $a$ reaching the same corner, which is why vector addition is commutative.

Fig 2The triangle law and the parallelogram law for a + b, the difference a − b drawn as a + (−b), and the scalar multiples 2a and −a.

VideoMr. Cioffi's MCV4U lesson on adding vectors geometrically by placing them tip to tail and finding the resultant.Watch on YouTube

Subtracting and scaling

Subtraction is addition of the opposite: $a - b = a + (- b)$ . Reverse $b$ and add it tip to tail. There is also a quick picture: with $a$ and $b$ drawn tail to tail, $a - b$ is the arrow from the head of $b$ to the head of $a$ . You can check it: starting with $b$ and then adding $a - b$ must land on $a$ . In a parallelogram with sides $a$ and $b$ , the two diagonals are $a + b$ and $a - b$ .

Multiplying by a scalar $k$ stretches the vector: $∣ k a ∣ = ∣ k ∣ ∣ a ∣$ . If $k > 0$ the direction is unchanged, if $k < 0$ it reverses, and $0 a = 0$ . Two non-zero vectors are COLLINEAR (parallel) exactly when one is a scalar multiple of the other. Dividing a non-zero vector by its own magnitude gives the UNIT VECTOR $\frac{a}{∣ a ∣}$ : length $1$ , same direction.

Loading interactive…

In the Add and subtract tab, drag the tips of a and b to see a + b or a − b drawn by the triangle or parallelogram construction, or scale a by k; the result's magnitude and direction update live.

VideoMr. Cioffi shows what multiplying a vector by a scalar does to its length and direction, including negative scalars that reverse it.Watch on YouTube

Magnitudes from the cosine law

∣ a + b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} + 2∣ a ∣∣ b ∣ cos θ ∣ a - b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} - 2∣ a ∣∣ b ∣ cos θ

θ is the angle between the vectors drawn tail to tail.

Why the plus sign? In the tip-to-tail triangle for $a + b$ , the interior angle where the vectors meet is $180° - θ$ , not $θ$ . The cosine law gives $∣ a ∣^{2} + ∣ b ∣^{2} - 2∣ a ∣∣ b ∣ cos (180° - θ),$ and $cos (180° - θ) = - cos θ$ flips the sign. For $a - b$ the triangle (tails together) contains $θ$ itself, so the usual minus sign stays. Once you have a magnitude, the sine law gives the direction.

Property	Statement
Commutative	$a + b = b + a$
Associative	$(a + b) + c = a + (b + c)$ and $(km) a = k (m a)$
Distributive	$k (a + b) = k a + k b$ and $(k + m) a = k a + m a$
Zero vector (identity)	$a + 0 = a$
Opposite (inverse)	$a + (- a) = 0$
Not commutative	$a - b = - (b - a)$ , so subtraction order matters

Properties of vector addition and scalar multiplication (k and m are scalars).

Simplify like algebra

Because these properties match the ones for numbers, you can expand brackets and collect like terms in

a

and

b

exactly as you would with

x

and

y

. The one thing you cannot do is divide by a vector:

\frac{a}{b}

has no meaning.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

∣ a ∣ = 5

∣ b ∣ = 8

, and the angle between

a

and

b

(tail to tail) is

60°

. Find

∣ a + b ∣

, the angle between

a + b

and

a

, and

∣ a - b ∣

Example 2

Simplify

3 (a - 2 b) - 2 (a - b) + b

Example 3

In parallelogram

O A BC

(vertices in order),

O A = a

and

OC = c

M

is the midpoint of

A B

and

N

is the midpoint of

BC

. Express

OB

A C

and

MN

in terms of

a

and

c

, and describe how

MN

relates to

A C

Where marks get lost

Adding magnitudes: $∣ a + b ∣ = ∣ a ∣ + ∣ b ∣$ only when the vectors point the same way (or one of them is $0$ ). In general $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ .
Using the $60°$ angle between the vectors directly in the cosine law for $a + b$ . In the tip-to-tail triangle the interior angle is $180° - 60° = 120°$ .
Drawing $a - b$ from the head of $a$ to the head of $b$ . That arrow is $b - a$ ; with tails together, $a - b$ points TO the head of $a$ .
Forgetting that a negative scalar reverses the direction: $- 2 a$ is twice as long as $a$ AND points the opposite way.
Joining tail to tail when using the triangle law. The triangle law needs head to tail; tail to tail is the parallelogram law, where the sum is the diagonal.

Flashcards 7

Triangle law of addition

Parallelogram law

a - b

∣ k a ∣

+ 3 more

Practice

∣ a ∣ = 3

∣ b ∣ = 4

, and the angle between them (tail to tail) is

120°

. Find

∣ a + b ∣

to 2 decimal places.

∣ u ∣ = 6

∣ v ∣ = 10

, and the angle between them (tail to tail) is

60°

. Find

∣ u - v ∣

to 2 decimal places.

∣ v ∣ = 4

, what is

∣ - 3 v ∣

Simplify

2 (a + 3 b) - 3 (a - b)

Match each property to the statement that illustrates it.

A. $(a + b) + c = a + (b + c)$
B. $a + (- a) = 0$
C. $a + b = b + a$
D. $k (a + b) = k a + k b$

Commutative property of addition

Associative property of addition

Distributive property over vector addition

Additive inverse

Which statement is FALSE for all non-zero vectors

a

and

b

with

a \neq = b

A parallelogram has sides

a

and

b

drawn from the same corner. Which pair of vectors runs along its two diagonals?

∣ a ∣ = 5

∣ b ∣ = 8

and the angle between them (tail to tail) is

60°

. Sort these magnitudes from smallest to largest.

1 $∣2 b ∣$
2 $∣2 a + b ∣$
3 $∣ a + b ∣$
4 $∣ a - b ∣$
5 $∣ b ∣$

Two non-zero vectors satisfy

b = - 4 a

. One word: are they collinear or non-collinear?

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