C. Geometry and Algebra of Vectors · Vectors in Two-Space and Three-Space

Properties of Vector Addition and Scalar Multiplication

C2.2About 65 minutes

Learning goals

I can state the commutative, associative and distributive properties and the zero, opposite and identity properties of vector operations.
I can investigate a property with examples, and disprove a false statement with a counterexample.
I can prove properties algebraically (with components) and geometrically (with diagrams).
I can simplify vector expressions and solve vector equations using the properties.

Words to know

commutative property
Order doesn't matter: 𝐮 + 𝐯 = 𝐯 + 𝐮.

associative property
Grouping doesn't matter: (𝐮 + 𝐯) + 𝐰 = 𝐮 + (𝐯 + 𝐰), and a(b𝐮) = (ab)𝐮.

distributive property
a(𝐮 + 𝐯) = a𝐮 + a𝐯 and (a + b)𝐮 = a𝐮 + b𝐮.

counterexample
One example that shows a general statement is false.

vector equation
An equation in which the unknown is a vector, such as 2𝐱 + 𝐮 = 𝐯.

Why rules matter

In algebra you rearrange 3(x + 2) − x without a second thought, because you trust rules like the distributive property. Can you do the same with vectors? Can you write 3(2𝐮 − 𝐯) − 2(𝐮 + 4𝐯) = 4𝐮 − 11𝐯? Yes, but only because vector addition and scalar multiplication obey a set of properties that we can check and prove. These properties are what make vectors part of linear algebra, a first-year university course in mathematics, engineering, computer science and economics.

Property	Statement (for all vectors 𝐮, 𝐯, 𝐰 and scalars a, b)
Commutative (addition)	𝐮 + 𝐯 = 𝐯 + 𝐮
Associative (addition)	(𝐮 + 𝐯) + 𝐰 = 𝐮 + (𝐯 + 𝐰)
Zero vector (identity)	𝐮 + 𝟎 = 𝐮
Opposite (inverse)	𝐮 + (−𝐮) = 𝟎
Associative (scalars)	a(b𝐮) = (ab)𝐮
Distributive over vector addition	a(𝐮 + 𝐯) = a𝐮 + a𝐯
Distributive over scalar addition	(a + b)𝐮 = a𝐮 + b𝐮
Scalar identity	1𝐮 = 𝐮

The properties of vector addition and scalar multiplication. They hold in two-space and three-space.

Investigating the properties

You can investigate a property by testing it on examples (by hand, with a spreadsheet, or with free graphing or geometry software). Examples can't prove a property is always true, but one counterexample proves that a statement is false.

To prove a property, use either

geometry: draw the vectors and use facts about parallelograms, triangles and similar figures; or
algebra: write the vectors in components and use the properties of real numbers, one component at a time.

Commutative property: going along 𝐮 then 𝐯, or along 𝐯 then 𝐮, ends at the same corner of the parallelogram.

Worked example

Example 1: Testing the distributive property

For 𝐮 = [2, −1, 3], 𝐯 = [0, 4, 1] and a = 3, check that a(𝐮 + 𝐯) = a𝐮 + a𝐯.

Left side: 𝐮 + 𝐯 = [2, 3, 4], so 3(𝐮 + 𝐯) = [6, 9, 12].
Right side: 3𝐮 = [6, −3, 9] and 3𝐯 = [0, 12, 3], so 3𝐮 + 3𝐯 = [6, 9, 12].
The sides agree for this example. (That supports the property but does not prove it.)

Both sides equal [6, 9, 12].

Worked example

Example 2: An algebraic proof

Prove that 𝐮 + 𝐯 = 𝐯 + 𝐮 for all vectors in three-space.

Let 𝐮 = [u₁, u₂, u₃] and 𝐯 = [v₁, v₂, v₃].
𝐮 + 𝐯 = [u₁ + v₁, u₂ + v₂, u₃ + v₃].
Addition of real numbers is commutative, so each component can be swapped: = [v₁ + u₁, v₂ + u₂, v₃ + u₃].
This is 𝐯 + 𝐮 by the definition of vector addition. ∎

Proved for every pair of vectors, because the argument used only letters, not particular numbers.

Associative property: adding 𝐮 + 𝐯 first and then 𝐰, or adding 𝐯 + 𝐰 first, gives the same resultant from the tail of 𝐮 to the head of 𝐰.

Worked example

Example 3: A geometric proof of the distributive property

Explain geometrically why 2(𝐮 + 𝐯) = 2𝐮 + 2𝐯.

Draw 𝐮 and then 𝐯 tip to tail: a triangle with third side 𝐮 + 𝐯.
Draw 2𝐮 and then 2𝐯 tip to tail from the same starting point. Each side is twice as long and parallel to the matching side of the first triangle.
Two sides in the ratio 2 : 1 with the same included angle make the triangles similar (SAS similarity), so the third side of the big triangle is parallel to 𝐮 + 𝐯 and twice as long.
That third side is 2𝐮 + 2𝐯, and it equals 2(𝐮 + 𝐯). The same argument works for any positive scalar, and reversing every arrow (a half-turn) handles negative scalars. ∎

2(𝐮 + 𝐯) = 2𝐮 + 2𝐯 because the triangles are similar.

The big triangle is the small triangle scaled by 2, so its third side 2(𝐮 + 𝐯) is the same as 2𝐮 + 2𝐯.

Worked example

Example 4: Simplifying with the properties

Simplify 3(2𝐮 − 𝐯) − 2(𝐮 + 4𝐯).

Distribute: 6𝐮 − 3𝐯 − 2𝐮 − 8𝐯.
Use commutativity and associativity to group like vectors: (6𝐮 − 2𝐮) + (−3𝐯 − 8𝐯).
Use (a + b)𝐮 = a𝐮 + b𝐮: 4𝐮 − 11𝐯.

4𝐮 − 11𝐯

Worked example

Example 5: Solving a vector equation

Solve 2𝐱 + 3𝐮 = 5𝐯 − 𝐱 for 𝐱. Then find 𝐱 if 𝐮 = [1, 2] and 𝐯 = [3, −3].

Add 𝐱 to both sides and subtract 3𝐮 from both sides: 3𝐱 = 5𝐯 − 3𝐮.
Multiply both sides by 1/3: 𝐱 = (5/3)𝐯 − 𝐮.
Substitute: (5/3)[3, −3] − [1, 2] = [5, −5] − [1, 2] = [4, −7].
Check: 2[4, −7] + 3[1, 2] = [11, −8], and 5[3, −3] − [4, −7] = [11, −8] ✓

𝐱 = (5/3)𝐯 − 𝐮 = [4, −7]

What does NOT work

Subtraction is not commutative: 𝐮 − 𝐯 = −(𝐯 − 𝐮), which is different from 𝐯 − 𝐮 unless 𝐮 = 𝐯. Subtraction is not associative either. And you cannot divide by a vector: an equation like 𝐮𝐱 = 𝐯 has no meaning here. You can only multiply or divide by scalars. (Two different ways to "multiply" vectors together appear in the unit Dot and Cross Products, and they follow their own rules.)

Just for fun

Vectors are very easy-going about addition: 𝐮 + 𝐯 or 𝐯 + 𝐮, they don't mind who goes first. Subtraction is another story. Ask for 𝐮 − 𝐯 and 𝐯 − 𝐮 and you get two arrows pointing in exactly opposite directions.

Hands-on activity

Dice-roll property checker

You need: two dice (or a free dice app), paper and a pencil.

Make vectors in three-space by rolling: for each component, roll one die and subtract 3, so components are −2 to 3. Make 𝐮, 𝐯 and 𝐰 this way. Roll once more (no subtraction) for a scalar a and again for b.
Check each property in the table for your vectors: compute the left side and the right side separately.
Now test two false statements: is 𝐮 − 𝐯 = 𝐯 − 𝐮? Is (𝐮 − 𝐯) − 𝐰 = 𝐮 − (𝐯 − 𝐰)?
Repeat with new rolls. Write a sentence explaining why examples can support a property, but one counterexample is enough to disprove a statement.

What to look for: all eight properties should check out every time. For the false statements, the two sides should differ. In fact 𝐮 − 𝐯 and 𝐯 − 𝐮 agree only if 𝐮 = 𝐯, and (𝐮 − 𝐯) − 𝐰 differs from 𝐮 − (𝐯 − 𝐰) by exactly 2𝐰, so they agree only if you rolled 𝐰 = 𝟎.

Your turn

Name the property: 𝐮 + (𝐯 + 𝐰) = (𝐮 + 𝐯) + 𝐰.

Name the property: 3(𝐮 + 𝐯) = 3𝐮 + 3𝐯.

Name the property: (2 + 5)𝐮 = 2𝐮 + 5𝐮.

Simplify 2(𝐮 + 3𝐯) − 4(𝐮 − 𝐯).

Solve 4𝐱 − 𝐮 = 2𝐱 + 3𝐮 for 𝐱.

Check the commutative property for 𝐮 = [1, 3] and 𝐯 = [−2, 5].

Is (𝐮 − 𝐯) − 𝐰 = 𝐮 − (𝐯 − 𝐰) always true? Give a counterexample or a reason.

More practice: warm-up

Simplify 𝐮 + 𝟎.

Simplify 𝐮 + (−𝐮).

Simplify 1𝐮.

Simplify 5(2𝐮).

Simplify −(𝐮 − 𝐯).

More practice: on your own

Simplify 3(𝐚 − 2𝐛) + 2(2𝐚 + 𝐛) − (𝐚 − 𝐛).

Solve 3(𝐱 + 𝐮) = 2(𝐱 − 𝐯) for 𝐱.

Prove (a + b)𝐮 = a𝐮 + b𝐮 for 𝐮 = [u₁, u₂].

𝐮 = [2, −1, 0] and 𝐯 = [1, 1, 3]. Find 𝐱 if 2𝐱 − 3𝐮 = 𝐯 + 𝐱.

Use components to show that |−3𝐮| = 3|𝐮| for 𝐮 = [a, b, c].

Challenge

Prove the associative property (𝐮 + 𝐯) + 𝐰 = 𝐮 + (𝐯 + 𝐰) for vectors in three-space.

Use the properties (not components) to prove that (−1)𝐮 = −𝐮.

Prove: if a𝐮 = 𝟎 and a ≠ 0, then 𝐮 = 𝟎.

Check your understanding

Quiz

Which property is shown by (𝐮 + 𝐯) + 𝐰 = 𝐮 + (𝐯 + 𝐰)?

Which property is shown by 4(𝐮 − 𝐯) = 4𝐮 − 4𝐯?

What is 2(3𝐮 + 𝐯) − (𝐮 − 𝐯)?

Which statement is NOT true for all vectors?

Solve 3𝐱 + 𝐮 = 𝐱 − 𝐯 for 𝐱.

|𝐮| = 4. What is |−2.5𝐮|?

For parents and teachers

Big idea (C2.2): vector addition and scalar multiplication follow the same kinds of rules as ordinary arithmetic (commutative, associative, distributive, zero and identity), so students can simplify vector expressions and solve vector equations with confidence. Subtraction is not commutative, and you can't divide by a vector.

How to help: The dice activity is quick and makes a good conversation about proof: "Does checking ten examples prove it? What would?" Ask your teen to say the name of the property they use at each step when simplifying.

Common difficulties:

Treating vectors like numbers and "dividing by 𝐮".
Sign errors when distributing a negative: −2(𝐮 − 4𝐯) = −2𝐮 + 8𝐯.
Thinking examples prove a property; a proof must use general components or a general diagram.
Mixing up the two distributive properties.