The Central Idea
If one vector represents one directed change and a second vector represents another directed change, their sum represents the combined change.
If
\[ \mathbf{u} \]occurs first and
\[ \mathbf{v} \]occurs second, then the total change is
\[ \boxed{\mathbf{u}+\mathbf{v}}. \]Geometrically, the vectors can be placed head to tail.
Head-to-Tail Addition
To add two vectors geometrically:
- Draw the first vector.
- Move the second vector without rotating or resizing it.
- Place the initial point of the second at the terminal point of the first.
- Draw the resultant from the initial point of the first to the terminal point of the second.
That resultant represents
\[ \boxed{\mathbf{u}+\mathbf{v}}. \]Why Vectors May Be Moved for Addition
In elementary vector geometry, vectors are usually treated as free vectors.
A free vector may be translated to another location without changing the vector, provided its magnitude and direction remain unchanged.
This is what allows us to arrange vectors head to tail.
Example: Two Successive Displacements
Suppose a person moves \(4\) units east and then \(3\) units north.
Let
\[ \mathbf{u}=\langle 4,0\rangle \]and
\[ \mathbf{v}=\langle 0,3\rangle. \]The combined displacement is
\[ \mathbf{u}+\mathbf{v} = \langle 4,3\rangle. \]The two-step trip is therefore represented by one resultant vector.
Component-Wise Vector Addition
If
\[ \mathbf{u}=\langle a,b\rangle \]and
\[ \mathbf{v}=\langle c,d\rangle, \]then add corresponding components:
\[ \boxed{ \mathbf{u}+\mathbf{v} = \langle a+c,\;b+d\rangle }. \]Horizontal changes combine with horizontal changes, and vertical changes combine with vertical changes.
Example: Add Two Vectors
Let
\[ \mathbf{u}=\langle 2,5\rangle \]and
\[ \mathbf{v}=\langle 4,-1\rangle. \]Then
\[ \mathbf{u}+\mathbf{v} = \langle 2+4,\;5+(-1)\rangle. \]Therefore,
\[ \boxed{ \mathbf{u}+\mathbf{v} = \langle 6,4\rangle }. \]Why Component Addition Works
Suppose \(\mathbf{u}\) contributes \(a\) horizontal units and \(b\) vertical units.
Suppose \(\mathbf{v}\) then contributes \(c\) horizontal units and \(d\) vertical units.
The total horizontal change is
\[ a+c, \]while the total vertical change is
\[ b+d. \]This produces
\[ \langle a+c,b+d\rangle. \]The Resultant Vector
A vector formed by combining two or more vectors is often called the resultant.
If
\[ \mathbf{R} = \mathbf{u}+\mathbf{v}, \]then \(\mathbf{R}\) summarizes the net directed effect of \(\mathbf{u}\) and \(\mathbf{v}\).
Resultant Does Not Mean Total Distance
Vector addition gives net directed change, not total path length.
If you travel \(5\) units east and then \(5\) units west, your total distance is
\[ 10, \]but your resultant displacement is
\[ \langle 5,0\rangle+\langle -5,0\rangle = \boxed{\langle 0,0\rangle}. \]The Parallelogram Rule
Two vectors can also be added by placing their initial points together.
Complete a parallelogram using the two vectors as adjacent sides.
The diagonal beginning at the common initial point represents
\[ \boxed{\mathbf{u}+\mathbf{v}}. \]This is called the parallelogram rule.
Head-to-Tail and Parallelogram Addition Agree
The head-to-tail rule and the parallelogram rule are not competing methods.
They are two geometric constructions of the same vector sum.
Component addition gives the same result algebraically.
Vector Addition Is Commutative
For vectors of the same dimension,
\[ \boxed{ \mathbf{u}+\mathbf{v} = \mathbf{v}+\mathbf{u} }. \]In component form,
\[ \langle a+c,b+d\rangle = \langle c+a,d+b\rangle \]because ordinary scalar addition is commutative.
Geometric Meaning of Commutativity
If two vectors are arranged head to tail in one order, their resultant is the same as when the order is reversed.
The intermediate path changes, but the overall displacement does not.
This distinction between path and resultant is important.
Vector Addition Is Associative
For compatible vectors,
\[ \boxed{ (\mathbf{u}+\mathbf{v})+\mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w}) }. \]Thus several vector additions may be grouped in different ways without changing the final resultant.
Adding Several Vectors
Suppose
\[ \mathbf{u}=\langle 2,1\rangle, \qquad \mathbf{v}=\langle -1,4\rangle, \qquad \mathbf{w}=\langle 3,-2\rangle. \]Then
\[ \mathbf{u}+\mathbf{v}+\mathbf{w} = \langle 2-1+3,\;1+4-2\rangle. \]Therefore,
\[ \boxed{ \mathbf{u}+\mathbf{v}+\mathbf{w} = \langle 4,3\rangle }. \]The Zero Vector Is the Additive Identity
For every vector \(\mathbf{v}\),
\[ \boxed{ \mathbf{v}+\mathbf{0} = \mathbf{v} }. \]In two dimensions,
\[ \langle a,b\rangle+\langle 0,0\rangle = \langle a,b\rangle. \]Opposite Vectors Are Additive Inverses
The opposite of
\[ \mathbf{v}=\langle a,b\rangle \]is
\[ -\mathbf{v} = \langle -a,-b\rangle. \]Their sum is
\[ \mathbf{v}+(-\mathbf{v}) = \langle 0,0\rangle. \]Therefore,
\[ \boxed{ \mathbf{v}+(-\mathbf{v}) = \mathbf{0} }. \]Vector Subtraction
Vector subtraction is defined using the additive inverse.
\[ \boxed{ \mathbf{u}-\mathbf{v} = \mathbf{u}+(-\mathbf{v}) }. \]So subtracting a vector means adding the same magnitude in the opposite direction.
Component-Wise Vector Subtraction
If
\[ \mathbf{u}=\langle a,b\rangle \]and
\[ \mathbf{v}=\langle c,d\rangle, \]then
\[ \boxed{ \mathbf{u}-\mathbf{v} = \langle a-c,\;b-d\rangle }. \]Example: Subtract Two Vectors
Let
\[ \mathbf{u}=\langle 7,2\rangle \]and
\[ \mathbf{v}=\langle 3,5\rangle. \]Then
\[ \mathbf{u}-\mathbf{v} = \langle 7-3,\;2-5\rangle. \]Therefore,
\[ \boxed{ \mathbf{u}-\mathbf{v} = \langle 4,-3\rangle }. \]Subtraction as Addition of an Opposite
Using the same vectors,
\[ \mathbf{u}=\langle 7,2\rangle, \qquad \mathbf{v}=\langle 3,5\rangle, \]first find
\[ -\mathbf{v} = \langle -3,-5\rangle. \]Then
\[ \mathbf{u}+(-\mathbf{v}) = \langle 7-3,\;2-5\rangle = \langle 4,-3\rangle. \]This confirms the subtraction rule.
Geometric Meaning of \(\mathbf{u}-\mathbf{v}\)
Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are drawn from the same initial point.
Then
\[ \mathbf{u}-\mathbf{v} \]can be represented by a vector pointing from the terminal point of \(\mathbf{v}\) to the terminal point of \(\mathbf{u}\).
Why the Difference Points from \(\mathbf{v}\) to \(\mathbf{u}\)
Rearrange
\[ \mathbf{u}-\mathbf{v} = \mathbf{d}. \]Adding \(\mathbf{v}\) to both sides gives
\[ \mathbf{u} = \mathbf{v}+\mathbf{d}. \]Therefore, \(\mathbf{d}\) is the vector that must be added to \(\mathbf{v}\) to reach \(\mathbf{u}\).
Position Vectors and Displacement
Suppose two points have position vectors
\[ \mathbf{p} \]and
\[ \mathbf{q}. \]The displacement from the point represented by \(\mathbf{p}\) to the point represented by \(\mathbf{q}\) is
\[ \boxed{ \mathbf{q}-\mathbf{p} }. \]This is another form of terminal minus initial.
Example: Displacement from Position Vectors
Let
\[ \mathbf{p}=\langle 2,1\rangle \]and
\[ \mathbf{q}=\langle 8,5\rangle. \]The displacement from \(\mathbf{p}\) to \(\mathbf{q}\) is
\[ \mathbf{q}-\mathbf{p} = \langle 8-2,\;5-1\rangle. \]Therefore,
\[ \boxed{\langle 6,4\rangle}. \]Connecting Point Notation and Vector Subtraction
If
\[ A=(x_1,y_1) \]and
\[ B=(x_2,y_2), \]their position vectors are
\[ \mathbf{a}=\langle x_1,y_1\rangle, \qquad \mathbf{b}=\langle x_2,y_2\rangle. \]Then
\[ \overrightarrow{AB} = \mathbf{b}-\mathbf{a} = \langle x_2-x_1,\;y_2-y_1\rangle. \]A Closed Vector Path
If several successive vectors bring us back to the starting point, their sum is the zero vector.
For example, if
\[ \mathbf{u}+\mathbf{v}+\mathbf{w} = \mathbf{0}, \]the three vectors form a closed head-to-tail path.
Example: Closed Triangle
Suppose
\[ \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA}. \]Traveling from \(A\) to \(B\), then \(B\) to \(C\), then \(C\) back to \(A\) produces no net displacement.
Therefore,
\[ \boxed{ \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \mathbf{0} }. \]The Triangle Relation
If you travel from \(A\) to \(B\) and then from \(B\) to \(C\), the total displacement is from \(A\) directly to \(C\).
\[ \boxed{ \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} }. \]This is the head-to-tail rule written using point notation.
Verifying the Triangle Relation Algebraically
Let
\[ A=(x_1,y_1),\quad B=(x_2,y_2),\quad C=(x_3,y_3). \]Then
\[ \overrightarrow{AB} = \langle x_2-x_1,\;y_2-y_1\rangle \]and
\[ \overrightarrow{BC} = \langle x_3-x_2,\;y_3-y_2\rangle. \]Adding gives
\[ \langle x_3-x_1,\;y_3-y_1\rangle = \overrightarrow{AC}. \]Adding Vectors in Three Dimensions
If
\[ \mathbf{u} = \langle a,b,c\rangle \]and
\[ \mathbf{v} = \langle d,e,f\rangle, \]then
\[ \boxed{ \mathbf{u}+\mathbf{v} = \langle a+d,\;b+e,\;c+f\rangle }. \]Subtracting Vectors in Three Dimensions
Using the same notation,
\[ \boxed{ \mathbf{u}-\mathbf{v} = \langle a-d,\;b-e,\;c-f\rangle }. \]The rule is unchanged: operate on corresponding components.
Example: Three-Dimensional Addition
Let
\[ \mathbf{u}=\langle 2,-1,4\rangle \]and
\[ \mathbf{v}=\langle 3,5,-2\rangle. \]Then
\[ \mathbf{u}+\mathbf{v} = \langle 2+3,\;-1+5,\;4-2\rangle. \]Therefore,
\[ \boxed{ \mathbf{u}+\mathbf{v} = \langle 5,4,2\rangle }. \]Example: Three-Dimensional Subtraction
With
\[ \mathbf{u}=\langle 2,-1,4\rangle, \qquad \mathbf{v}=\langle 3,5,-2\rangle, \]we have
\[ \mathbf{u}-\mathbf{v} = \langle 2-3,\;-1-5,\;4-(-2)\rangle. \]Thus,
\[ \boxed{ \mathbf{u}-\mathbf{v} = \langle -1,-6,6\rangle }. \]Vector Addition in \(\mathbf{i},\mathbf{j}\) Form
Suppose
\[ \mathbf{u}=2\mathbf{i}+3\mathbf{j} \]and
\[ \mathbf{v}=5\mathbf{i}-\mathbf{j}. \]Combine like unit-vector terms:
\[ \mathbf{u}+\mathbf{v} = (2+5)\mathbf{i} +(3-1)\mathbf{j}. \]Therefore,
\[ \boxed{ \mathbf{u}+\mathbf{v} = 7\mathbf{i}+2\mathbf{j} }. \]Vector Subtraction in \(\mathbf{i},\mathbf{j}\) Form
If
\[ \mathbf{u}=4\mathbf{i}+7\mathbf{j} \]and
\[ \mathbf{v}=\mathbf{i}+3\mathbf{j}, \]then
\[ \mathbf{u}-\mathbf{v} = (4-1)\mathbf{i} +(7-3)\mathbf{j}. \]Hence,
\[ \boxed{ \mathbf{u}-\mathbf{v} = 3\mathbf{i}+4\mathbf{j} }. \]Net Change over Several Stages
Vectors are particularly useful when motion or change occurs in several stages.
If the stage vectors are
\[ \mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n, \]then the net change is
\[ \boxed{ \mathbf{R} = \mathbf{v}_1+\mathbf{v}_2+\cdots+\mathbf{v}_n }. \]Example: Net Motion
Suppose a robot moves according to
\[ \langle 3,2\rangle, \qquad \langle -1,4\rangle, \qquad \langle 5,-3\rangle. \]The net displacement is
\[ \langle 3-1+5,\;2+4-3\rangle. \]Therefore,
\[ \boxed{\langle 7,3\rangle}. \]Magnitude of a Resultant
Once vectors are added, the magnitude of the resultant can be found using the ordinary magnitude formula.
If
\[ \mathbf{R}=\langle a,b\rangle, \]then
\[ \boxed{ \|\mathbf{R}\| = \sqrt{a^2+b^2} }. \]Example: Magnitude of a Resultant
Suppose
\[ \mathbf{u}=\langle 4,0\rangle \]and
\[ \mathbf{v}=\langle 0,3\rangle. \]Their resultant is
\[ \mathbf{R} = \langle 4,3\rangle. \]Its magnitude is
\[ \|\mathbf{R}\| = \sqrt{4^2+3^2} = \boxed{5}. \]The Magnitudes Usually Do Not Simply Add
In general,
\[ \|\mathbf{u}+\mathbf{v}\| \neq \|\mathbf{u}\|+\|\mathbf{v}\|. \]Direction matters.
For perpendicular vectors of magnitudes \(4\) and \(3\), the resultant magnitude is \(5\), not \(7\).
When Magnitudes Do Add
If two nonzero vectors point in exactly the same direction, then the magnitude of their sum equals the sum of their magnitudes.
For example,
\[ \langle 2,0\rangle+\langle 5,0\rangle = \langle 7,0\rangle. \]Thus,
\[ 2+5=7. \]Partial Cancellation
Opposing component directions can reduce a resultant.
For example,
\[ \langle 8,3\rangle+\langle -5,2\rangle = \langle 3,5\rangle. \]The horizontal components partially cancel because they point in opposite directions.
Complete Cancellation
A vector and its opposite cancel completely:
\[ \mathbf{v}+(-\mathbf{v}) = \mathbf{0}. \]For example,
\[ \langle 6,-2\rangle + \langle -6,2\rangle = \boxed{\langle 0,0\rangle}. \]Vector Subtraction Is Not Commutative
In general,
\[ \boxed{ \mathbf{u}-\mathbf{v} \neq \mathbf{v}-\mathbf{u} }. \]In fact,
\[ \mathbf{v}-\mathbf{u} = -(\mathbf{u}-\mathbf{v}). \]Example: Order Matters in Subtraction
Let
\[ \mathbf{u}=\langle 5,2\rangle, \qquad \mathbf{v}=\langle 1,4\rangle. \]Then
\[ \mathbf{u}-\mathbf{v} = \langle 4,-2\rangle, \]while
\[ \mathbf{v}-\mathbf{u} = \langle -4,2\rangle. \]These are opposite vectors.
Common Error: Adding All Components Together
If
\[ \mathbf{u}=\langle 2,3\rangle \]and
\[ \mathbf{v}=\langle 4,5\rangle, \]do not write
\[ 2+3+4+5=14. \]Vectors are added component by component:
\[ \boxed{ \mathbf{u}+\mathbf{v} = \langle 6,8\rangle }. \]Common Error: Mixing Component Positions
The first component must combine with the first component.
The second must combine with the second.
For
\[ \langle a,b\rangle+\langle c,d\rangle, \]the correct result is
\[ \boxed{\langle a+c,b+d\rangle}, \]not \(\langle a+d,b+c\rangle\).
Common Error: Forgetting to Negate Every Component
If
\[ \mathbf{v}=\langle 3,-4\rangle, \]then
\[ -\mathbf{v} = \boxed{\langle -3,4\rangle}. \]Every component changes sign.
Common Error: Treating Subtraction as Reordering
Vector subtraction does not mean placing the smaller-looking vector first.
The order is determined by the expression:
\[ \mathbf{u}-\mathbf{v} = \mathbf{u}+(-\mathbf{v}). \]Reversing the order produces the opposite result.
Common Error: Adding Magnitudes Instead of Vectors
Knowing
\[ \|\mathbf{u}\|=4 \]and
\[ \|\mathbf{v}\|=3 \]is not enough to conclude that
\[ \|\mathbf{u}+\mathbf{v}\|=7. \]The directions of the vectors must also be considered.
Common Error: Confusing Path Length with Resultant Magnitude
Several movements may accumulate a large total distance while producing a small net displacement.
Vector addition tracks net directed change.
Distance traveled and magnitude of displacement are different quantities.
A Reliable Procedure for Vector Addition
- Verify that the vectors have the same dimension.
- Line up corresponding components.
- Add first components.
- Add second components.
- Continue component by component in higher dimensions.
- Interpret the resulting vector geometrically when useful.
- Find its magnitude or direction only if the problem asks for them.
A Reliable Procedure for Vector Subtraction
- Preserve the order of the vectors.
- Replace subtraction with addition of the opposite if helpful.
- Change the sign of every component of the vector being subtracted.
- Add corresponding components.
- Check that reversing the subtraction would reverse the result.
A Reliable Procedure for Net Displacement
- Represent each stage of motion as a vector.
- Choose one consistent coordinate orientation.
- Add all horizontal components.
- Add all vertical components.
- Include additional components in higher dimensions.
- Write the resultant vector.
- Compute magnitude or direction if required.
Example: Full Addition Workflow
Let
\[ \mathbf{u}=\langle -2,6\rangle, \qquad \mathbf{v}=\langle 5,-1\rangle. \]Add corresponding components:
\[ \mathbf{u}+\mathbf{v} = \langle -2+5,\;6-1\rangle. \]Therefore,
\[ \boxed{ \mathbf{u}+\mathbf{v} = \langle 3,5\rangle }. \]The resultant points right and upward.
Example: Full Subtraction Workflow
Let
\[ \mathbf{u}=\langle 8,-3\rangle, \qquad \mathbf{v}=\langle 2,4\rangle. \]First find the opposite:
\[ -\mathbf{v} = \langle -2,-4\rangle. \]Then add:
\[ \mathbf{u}-\mathbf{v} = \langle 8,-3\rangle + \langle -2,-4\rangle. \]Therefore,
\[ \boxed{ \mathbf{u}-\mathbf{v} = \langle 6,-7\rangle }. \]Practice 1: Add Two Vectors
Find
\[ \langle 3,7\rangle+\langle 5,-2\rangle. \]Practice 2: Subtract Two Vectors
Find
\[ \langle 9,4\rangle-\langle 2,6\rangle. \]Practice 3: Find an Opposite Vector
Find the opposite of
\[ \mathbf{v}=\langle -4,8\rangle. \]Practice 4: Add Three Vectors
Find
\[ \langle 1,2\rangle + \langle -3,5\rangle + \langle 4,-1\rangle. \]Practice 5: Find a Displacement
A point moves from
\[ A=(-2,3) \]to
\[ B=(5,-1). \]Find \(\overrightarrow{AB}\).
Practice 6: Three-Dimensional Addition
Find
\[ \langle 2,-1,5\rangle + \langle 4,3,-2\rangle. \]Practice 7: Three-Dimensional Subtraction
Find
\[ \langle 7,0,-3\rangle - \langle 2,5,4\rangle. \]Practice 8: Closed Path
If
\[ \mathbf{u}=\langle 3,1\rangle \]and
\[ \mathbf{v}=\langle -1,4\rangle, \]find a vector \(\mathbf{w}\) such that
\[ \mathbf{u}+\mathbf{v}+\mathbf{w} = \mathbf{0}. \]Practice 9: Resultant Magnitude
Let
\[ \mathbf{u}=\langle 6,0\rangle, \qquad \mathbf{v}=\langle 0,8\rangle. \]Find the magnitude of
\[ \mathbf{u}+\mathbf{v}. \]Practice 10: Identify the Error
A student writes
\[ \langle 5,-2\rangle - \langle 3,4\rangle = \langle 2,2\rangle. \]Identify and correct the error.
Practice Answers
Practice 1
\[ \langle 3,7\rangle+\langle 5,-2\rangle = \boxed{\langle 8,5\rangle}. \]Practice 2
\[ \langle 9,4\rangle-\langle 2,6\rangle = \boxed{\langle 7,-2\rangle}. \]Practice 3
\[ -\mathbf{v} = \boxed{\langle 4,-8\rangle}. \]Practice 4
\[ \langle 1-3+4,\;2+5-1\rangle = \boxed{\langle 2,6\rangle}. \]Practice 5
\[ \overrightarrow{AB} = \langle 5-(-2),\;-1-3\rangle = \boxed{\langle 7,-4\rangle}. \]Practice 6
\[ \langle 2,-1,5\rangle + \langle 4,3,-2\rangle = \boxed{\langle 6,2,3\rangle}. \]Practice 7
\[ \langle 7,0,-3\rangle - \langle 2,5,4\rangle = \boxed{\langle 5,-5,-7\rangle}. \]Practice 8
First,
\[ \mathbf{u}+\mathbf{v} = \langle 3-1,\;1+4\rangle = \langle 2,5\rangle. \]Therefore,
\[ \boxed{ \mathbf{w} = \langle -2,-5\rangle }. \]Practice 9
\[ \mathbf{u}+\mathbf{v} = \langle 6,8\rangle. \]Its magnitude is
\[ \sqrt{6^2+8^2} = \sqrt{100} = \boxed{10}. \]Practice 10
The second component was subtracted incorrectly.
\[ -2-4=-6, \]not \(2\).
Therefore,
\[ \boxed{ \langle 5,-2\rangle - \langle 3,4\rangle = \langle 2,-6\rangle }. \]What to Remember
- Vector addition combines directed changes.
- Geometrically, vectors may be added head to tail.
- The parallelogram rule gives the same sum.
- Components are added component by component.
- Vector addition is commutative and associative.
- The zero vector is the additive identity.
- The opposite vector is the additive inverse.
- Vector subtraction means adding the opposite.
- Subtraction is not commutative.
- Resultant displacement is not generally the same as total distance traveled.
- \(\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}\).
Where We Go Next
Vector addition combines directed quantities, while subtraction compares them through opposites. The next step is to change the size and possibly the direction of a vector by multiplying it by a scalar.
The next lesson, Scalar Multiplication and Linear Combinations, develops vector scaling, parallel and opposite directions, weighted sums, and the idea of building new vectors from combinations of existing ones.