Adding Vectors by Components

Adding Coordinate Changes

When a movement has several stages, its total change in position depends on the changes in each coordinate. If one part of a journey changes the $x$-coordinate by $3$ and the next changes it by $-1$, the total change in $x$ is $3+(-1)=2$. The same accounting applies to the $y$- and $z$-coordinates.

A three-dimensional vector records three signed components, one for each coordinate direction. Adding vectors means combining the changes in matching coordinates. The components must stay in their established order: first coordinate with first coordinate, second with second, and third with third.

Definition: If $\mathbf{u}=(u_1,u_2,u_3)$ and $\mathbf{v}=(v_1,v_2,v_3)$ are vectors in three-dimensional space, their sum is the vector $$ \mathbf{u}+\mathbf{v}=(u_1+v_1,\ u_2+v_2,\ u_3+v_3). $$ Thus each component of the sum is obtained by adding the corresponding components of the two vectors.

The result is still a three-dimensional vector: it has one component in each coordinate position. This rule does not combine unlike coordinates. For instance, the first component of the sum comes from the two first components, not from a first component and a second component.

Why Components Add This Way

The component rule follows from how coordinate changes accumulate. Suppose a first displacement changes a point’s coordinates by $(u_1,u_2,u_3)$. A second displacement then changes them by $(v_1,v_2,v_3)$. The first coordinate changes by $u_1$, then by $v_1$, so its total change is $u_1+v_1$. The second coordinate changes by $u_2$, then by $v_2$, giving $u_2+v_2$. The third coordinate changes by $u_3$, then by $v_3$, giving $u_3+v_3$.

Record the first vector’s components.
Write its three coordinate changes in order: $u_1,u_2,u_3$.

Record the second vector’s components.
Keep the same coordinate order: $v_1,v_2,v_3$.

Combine changes coordinate by coordinate.
The total changes are $u_1+v_1,\ u_2+v_2,\ u_3+v_3$.

This also justifies the rule algebraically: if the initial point is $(x,y,z)$, after the first displacement it is $(x+u_1,y+u_2,z+u_3)$. After the second displacement it is

(x+u_1+v_1,\ y+u_2+v_2,\ z+u_3+v_3).

Relative to the initial point, the total coordinate changes are therefore $(u_1+v_1,u_2+v_2,u_3+v_3)$, exactly the componentwise sum. The rule expresses ordinary addition of coordinate changes, carried out separately for each coordinate.

A Reliable Calculation Method

To add vectors, first write them so their components line up. Then add the numbers in each coordinate position, paying attention to signs, and place the three results in the same order. A useful habit is to show the three additions explicitly rather than trying to do the calculation mentally.

Worked Example: Adding Two Vectors with Signed Components

Let $\mathbf{u}=(3,-2,5)$ and $\mathbf{v}=(-1,4,2)$. Find $\mathbf{u}+\mathbf{v}$.

Match the first components, then the second, then the third:

\begin{aligned} \mathbf{u}+\mathbf{v} &=(3+(-1),\ -2+4,\ 5+2)\\ &=(2,\ 2,\ 7). \end{aligned}

The component calculations are $3+(-1)=2$, $-2+4=2$, and $5+2=7$. Thus $\mathbf{u}+\mathbf{v}=(2,2,7)$.

Check: The first output component is the sum of $3$ and $-1$, the second is the sum of $-2$ and $4$, and the third is the sum of $5$ and $2$. The resulting components are $2,2,7$, respectively, so each coordinate agrees with the rule.

Negative components do not change the procedure. They are signed numbers, so use the ordinary rules of arithmetic when adding them. For example, $-2+4=2$, while $4+(-7)=-3$. Do not remove a negative sign just because the component is part of a vector.

Worked Example: Adding Three Vectors

Let $\mathbf{a}=(2,1,-3)$, $\mathbf{b}=(-5,2,4)$, and $\mathbf{c}=(1,-4,2)$. Find their sum by adding matching components.

First combine $\mathbf{a}$ and $\mathbf{b}$:

\mathbf{a}+\mathbf{b} =(2+(-5),\ 1+2,\ -3+4) =(-3,\ 3,\ 1).

Now add $\mathbf{c}$, matching its components with the components of the intermediate result:

\begin{aligned} (\mathbf{a}+\mathbf{b})+\mathbf{c} &=(-3+1,\ 3+(-4),\ 1+2)\\ &=(-2,\ -1,\ 3). \end{aligned}

Thus the total is $(-2,-1,3)$. Equivalently, combine all inputs by coordinate: $2+(-5)+1=-2$, $1+2+(-4)=-1$, and $-3+4+2=3$.

Check: The first components total $-2$, the second components total $-1$, and the third components total $3$. These are precisely the three components of the result.

Interpreting a Sum as a Total Displacement

A vector sum can represent the net change produced by consecutive displacements. The starting point is not included in the sum of the displacement vectors; it is used afterward to find the final point. This distinction helps prevent confusing a point’s coordinates with a vector’s components.

Worked Example: Combining Displacements from a Starting Point

A point begins at $P=(1,-2,0)$. It undergoes a displacement $\mathbf{u}=(4,3,-1)$, followed by a displacement $\mathbf{v}=(-2,1,5)$. Find the total displacement and the final point.

Add the displacement vectors component by component:

\begin{aligned} \mathbf{u}+\mathbf{v} &=(4+(-2),\ 3+1,\ -1+5)\\ &=(2,\ 4,\ 4). \end{aligned}

The total displacement is $(2,4,4)$. Add this change to the starting point’s coordinates to get the final point:

(1,-2,0)+(2,4,4) =(1+2,\ -2+4,\ 0+4) =(3,2,4).

Check: After the first displacement, the point is $(1+4,-2+3,0+(-1))=(5,1,-1)$. After the second, it is $(5+(-2),1+1,-1+5)=(3,2,4)$, matching the final point found using the total displacement. The coordinate changes from $P$ to $(3,2,4)$ are $3-1=2$, $2-(-2)=4$, and $4-0=4$, which recover $(2,4,4)$.

Conditions and Common Pitfalls

The component formula applies when the vectors are expressed in the same three-dimensional coordinate system and their components are listed in the same order. If one vector’s entries are ordered as $x,y,z$ and another’s as $z,x,y$, matching entries by position would combine different coordinates. Put both vectors into the same order before adding.

Key takeaway: Add corresponding components, preserve their coordinate order, and treat negative components as signed numbers. The sum has three components because each vector has three.

A common mistake is to add components that do not correspond, or to ignore a negative sign. Here is an incorrect calculation for $(3,-2,5)+(-1,4,2)$:

(3,-2,5)+(-1,4,2) \ \overset{\text{wrong}}{=}\ (3+4,\ -2+2,\ 5+(-1)) =(7,0,4).

This calculation takes the second component of the second vector for the first addition, and shifts the other pairings as well. The entries must be paired by their positions. The correction is

(3+(-1),\ -2+4,\ 5+2)=(2,2,7).

Another error is to write only the numerical totals without retaining their coordinate positions. For instance, finding $2,2,7$ is not enough unless the result is recorded as the vector $(2,2,7)$, with the first, second, and third components in order. The order carries meaning.

Using the Rule Efficiently

For two vectors, the calculation can be displayed in aligned rows:

\begin{aligned} \mathbf{u}&=(u_1,\ u_2,\ u_3),\\ \mathbf{v}&=(v_1,\ v_2,\ v_3),\\ \mathbf{u}+\mathbf{v}&=(u_1+v_1,\ u_2+v_2,\ u_3+v_3). \end{aligned}

This layout makes it easier to see whether every component has been paired correctly. For a sum of more than two vectors, continue the same coordinate-by-coordinate accounting. Add all first components for the first entry, all second components for the second entry, and all third components for the third entry.

A quick check is to revisit each coordinate separately. Ask whether the first answer is the sum of only the first inputs, whether the second answer is the sum of only the second inputs, and whether the third answer is the sum of only the third inputs. This catches many sign and alignment errors before they carry into a later calculation.

Check Your Understanding

For each question, keep the components in their stated order and show the arithmetic in each coordinate.

Find $(2,-3,1)+(5,4,-2)$.
Find $(-4,1,6)+(3,-5,2)$, showing how the negative components are handled.
Compute the sum of $(1,2,-1)$, $(-3,0,4)$, and $(2,-5,1)$.
A point at $(0,1,-2)$ undergoes displacements $(3,-2,4)$ and $(-1,5,0)$. What is the total displacement and the final point?
Explain why adding the first component of one vector to the second component of another does not follow the componentwise addition rule.

What You'll Learn