Kuzee
Vector Algebra and Analytic Geometry of Space

Vector Addition

The Definition of Addition

Vector addition

Let a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3) and b⃗=(b1,b2,b3)\vec{b} = (b_1, b_2, b_3). Then:

a⃗+b⃗=(a1+b1,a2+b2,a3+b3)\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)
Geometric interpretation

Vector addition can be represented geometrically by the parallelogram rule or by the triangle rule.

The Geometric Meaning

The Parallelogram Rule

Translate the two vectors a⃗\vec{a} and b⃗\vec{b} so that they have a common starting point, build a parallelogram with these two vectors as adjacent sides, and the diagonal is the vector a⃗+b⃗\vec{a} + \vec{b}.

The Triangle Rule

Move the starting point of the vector b⃗\vec{b} to the endpoint of the vector a⃗\vec{a}; then the vector from the starting point of a⃗\vec{a} to the endpoint of b⃗\vec{b} is a⃗+b⃗\vec{a} + \vec{b}.

Properties of Addition

Properties of vector addition

The commutative law:

a⃗+b⃗=b⃗+a⃗\vec{a} + \vec{b} = \vec{b} + \vec{a}

The associative law:

(a⃗+b⃗)+c⃗=a⃗+(b⃗+c⃗)(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})

The zero vector:

a⃗+0⃗=a⃗\vec{a} + \vec{0} = \vec{a}

Worked Examples

Combining Displacements

If an object first moves along the vector a⃗\vec{a} and then along the vector b⃗\vec{b}, the total displacement is a⃗+b⃗\vec{a} + \vec{b}.

Combining Forces

When several forces act at the same time, the resultant force is the vector sum of the individual force vectors.

Exercises

Exercise 1

Given a⃗=(2,3,1)\vec{a} = (2, 3, 1) and b⃗=(1,−1,2)\vec{b} = (1, -1, 2), find a⃗+b⃗\vec{a} + \vec{b}.

vector additioncoordinate computation

Approach:

Use the coordinate rule for vector addition.

Steps:

  1. a⃗+b⃗=(2+1,3+(−1),1+2)=(3,2,3)\vec{a} + \vec{b} = (2+1, 3+(-1), 1+2) = (3, 2, 3)

Answer: a⃗+b⃗=(3,2,3)\vec{a} + \vec{b} = (3, 2, 3)

Exercise 2

Let a⃗=(1,−2,3)\vec{a} = (1, -2, 3) and b⃗=(4,5,−6)\vec{b} = (4, 5, -6). Verify that a⃗+b⃗=b⃗+a⃗\vec{a} + \vec{b} = \vec{b} + \vec{a}.

the commutative lawvector additioncoordinate computation

Approach:

Compute the sum in both orders and compare the results; the commutative law holds because addition of the components is itself commutative.

Steps:

  1. a⃗+b⃗=(1+4,−2+5,3+(−6))=(5,3,−3)\vec{a} + \vec{b} = (1+4, -2+5, 3+(-6)) = (5, 3, -3)
  2. b⃗+a⃗=(4+1,5+(−2),−6+3)=(5,3,−3)\vec{b} + \vec{a} = (4+1, 5+(-2), -6+3) = (5, 3, -3)
  3. The two results are identical, which confirms the commutative law

Answer: Both orders give (5,3,−3)(5, 3, -3), so a⃗+b⃗=b⃗+a⃗\vec{a} + \vec{b} = \vec{b} + \vec{a}.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
a⃗+b⃗\vec{a} + \vec{b}Vector operationa plus bVector addition
0⃗\vec{0}Vectorzero vectorThe zero vector
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