What Are Coinitial Vectors?
Coinitial vectors are two or more vectors that have the same initial point, the point where each vector begins. A vector has two ends: the starting point (the tail, also called the initial point) and the arrow tip (the head, or terminal point). When several vectors share the same tail, they are coinitial, regardless of the direction each one points or how long it is.
Consider a point $O$ with three arrows drawn out from it to points $A$, $B$, and $C$. The vectors $\vec{OA}$, $\vec{OB}$, and $\vec{OC}$ are coinitial because all three begin at $O$. Their heads land in three different places, and that is allowed. Only the shared start matters.
Two facts follow straight from the definition:
The endpoints can be anywhere. Coinitial vectors do not need to reach the same terminal point, and usually they do not.
Direction and length are free. Coinitial vectors can point any way and be any size. They can even be parallel or crossing, and it changes nothing about whether they are coinitial.
For the wider family these belong to, see types of vectors and the general idea of a vector.
How Do You Identify Coinitial Vectors In A Diagram?
To identify coinitial vectors, look only at where each arrow begins. Group the vectors by their tails. Any two or more that start at the same point are coinitial with each other.
Take a square $ABCD$ with its diagonals drawn in. Focus on the vectors that start at corner $A$: $\vec{AB}$ (along the top edge), $\vec{AD}$ (down the side), and $\vec{AC}$ (along the diagonal). All three begin at $A$, so $\vec{AB}$, $\vec{AD}$, and $\vec{AC}$ are coinitial. The vector $\vec{BC}$ is not part of this group, because it starts at $B$, not $A$.
A quick test: cover the arrowheads with your hand. If the remaining tails sit on top of each other at one point, the vectors are coinitial.
What Is The Difference Between Coinitial And Collinear Vectors?
This is the single most common mix-up, and the two ideas are checking completely different things.
Coinitial is about the starting point. Do the vectors share a tail?
Collinear is about direction. Do the vectors lie along the same line, meaning each is a scalar multiple of the other?
A vector can be one, both, or neither. Two arrows from point $O$ pointing in different directions are coinitial but not collinear. Two parallel arrows starting at different points are collinear but not coinitial. Two arrows from $O$ pointing the exact same way are both.
$$\vec{OA} = \begin{bmatrix} 2 \ 1 \end{bmatrix}, \quad \vec{OB} = \begin{bmatrix} 4 \ 2 \end{bmatrix}, \quad \vec{OC} = \begin{bmatrix} 1 \ 3 \end{bmatrix}$$
Here $\vec{OA}$, $\vec{OB}$, and $\vec{OC}$ all start at $O$, so all three are coinitial. Only $\vec{OA}$ and $\vec{OB}$ are also collinear, because $\vec{OB} = 2,\vec{OA}$ (one is a scalar multiple of the other). The vector $\vec{OC}$ is not a multiple of either, so it is coinitial with them but not collinear.
How Do Coinitial Vectors Compare To Equal And Coterminous Vectors?
Coinitial is one label in a family of vector classifications, and each label tests a different feature. Comparing them side by side is the fastest way to keep them apart.
Table: What each vector classification actually checks.
Type | What must match | Endpoints? | Direction/length? |
|---|---|---|---|
Coinitial | Same initial point (tail) | Can differ | Can differ |
Collinear | Same line (scalar multiples) | Can differ | Same or opposite direction |
Equal | Same magnitude and same direction | Free to sit anywhere | Both must match |
Coterminous | Same terminal point (head) | Must share the head | Can differ |
Read down the "what must match" column and the difference is clear. Coinitial vectors share the tail; coterminous vectors share the head; equal vectors match in both size and direction but can be drawn anywhere; collinear vectors follow one line. Note that equal vectors can also happen to be coinitial (if two identical arrows are drawn from the same point they land on top of each other), but the two labels are not the same test. For a related idea, see the negative of a vector, which shares magnitude but reverses direction.
Why Do Coinitial Vectors Matter?
Coinitial vectors are not just a term to memorise for an exam. The arrangement is the setup that two of the most important rules in vector math are built on.
The parallelogram law of addition. To add two vectors with the parallelogram law, you draw them from the same starting point, complete the parallelogram, and the diagonal from that shared point is the sum. The law only works when the two vectors are coinitial. That shared tail is not a detail, it is the whole construction.
Position vectors. Every position vector is measured from one fixed origin. A whole diagram of position vectors is, by definition, a set of coinitial vectors all starting at that origin. This is how points in space get turned into numbers.
Resolving forces. When several forces act on one object, physicists draw them all from the single point where they are applied. That single point makes them coinitial, and only then can the forces be combined into one resultant.
So the reason the label exists is practical. A lot of vector work requires the vectors to share a start before the method will run, and "coinitial" is the one word that says the setup is ready. The addition of vectors leans on this arrangement directly.
Who Discovered Vectors And Named Them?
The arrows are old, but the word "vector" and the algebra around it are surprisingly recent, and they came from one stubborn walk along a canal.
Two more mathematicians turned Hamilton's start into the vectors used today:
Josiah Willard Gibbs (1839–1903, United States) stripped the quaternion system down into the clean vector algebra, with its dot and cross products, that students learn now.
Oliver Heaviside (1850–1925, England) independently shaped the same modern vector notation while working on electromagnetism, making it the standard language of physics.
Where Are Coinitial Vectors Used In The Real World?
The shared-start arrangement shows up wherever several quantities act from one place at once.
Engineering and physics: a free-body diagram draws every force on an object from the single point of application, a coinitial set that is then added into one resultant force.
Navigation and aviation: an aircraft's heading, the wind, and the resulting ground track are drawn from one point to work out the true path across the ground.
Computer graphics and animation: a character's joint (a shoulder, say) is one origin from which several direction arrows are measured to place and move the connected limb.
Robotics: a robot arm's motions are planned as vectors from a shared joint origin, so the controller can combine them predictably.
Sports analysis: the forces a swimmer or rower applies are modelled from a single point on the body to find the net push through the water.
One small idea, several arrows from one point, is the starting frame for force analysis, flight paths, and animation rigs alike. Mathematics reuses the same simple setup across fields that look nothing like each other.
What Are The Most Common Coinitial Vector Mistakes?
These three errors account for most lost marks on vector-classification questions, and each is a case of testing the wrong feature.
Confusing coinitial with collinear.
Where it slips in:
A student sees two arrows starting at the same point and calls them collinear, or sees two parallel arrows and calls them coinitial.
Don't do this:
Do not treat "same start" and "same line" as the same condition. They are independent.
The correct way:
Check the tails for coinitial and the direction for collinear. Vectors from one point in different directions are coinitial but not collinear; parallel vectors from different points are collinear but not coinitial.
Assuming coinitial vectors must have equal length or direction.
Where it slips in:
A student decides a set is not coinitial because the arrows are different lengths or point different ways.
Don't do this:
Do not add conditions the definition never had. Coinitial says nothing about size or direction.
The correct way:
Judge only the starting point. If the tails coincide, the vectors are coinitial even when every arrow has a different length and a different direction.
Confusing coinitial with coterminous.
Where it slips in:
A student mixes up the two "co-" words and checks the arrowheads instead of the tails, or the other way round.
Don't do this:
Do not test the terminal points for coinitial. Coterminous vectors share the head; coinitial vectors share the tail.
The correct way:
Remember that "initial" means start. Coinitial vectors meet at the beginning; coterminous vectors meet at the end. The vectors $\vec{AC}$ and $\vec{BC}$ are coterminous (both end at $C$), while $\vec{CA}$ and $\vec{CB}$ are coinitial (both start at $C$).
Practice Problems On Coinitial Vectors
Use the diagram idea of a point $O$ with arrows to points $A$, $B$, $C$, and a separate point $P$ with an arrow to $Q$. Answers follow each line.
From the vectors $\vec{OA}$, $\vec{OB}$, $\vec{OC}$, and $\vec{PQ}$, which are coinitial?
(Answer: $\vec{OA}$, $\vec{OB}$, and $\vec{OC}$, because all three start at $O$. $\vec{PQ}$ starts at $P$, so it is not in the group.)In square $ABCD$, name three coinitial vectors starting at vertex $B$.
(Answer: $\vec{BA}$, $\vec{BC}$, and $\vec{BD}$, all begin at $B$.)Given $\vec{OA} = \begin{bmatrix} 3 \ 0 \end{bmatrix}$ and $\vec{OB} = \begin{bmatrix} 6 \ 0 \end{bmatrix}$, are they coinitial, collinear, or both?
(Answer: both, they share the start $O$ and $\vec{OB} = 2,\vec{OA}$ lies on the same line.)True or false: coinitial vectors must have the same magnitude.
(Answer: false, only the initial point must match.)Vectors $\vec{XY}$ and $\vec{XZ}$ start at $X$; vector $\vec{WZ}$ ends at $Z$. Which pair is coinitial and which pair is coterminous?
(Answer: $\vec{XY}$ and $\vec{XZ}$ are coinitial (shared start $X$); $\vec{XZ}$ and $\vec{WZ}$ are coterminous (shared end $Z$).)Can two equal vectors be coinitial?(Answer: yes, if two equal vectors are drawn from the same point they coincide exactly, so they are coinitial as well as equal, but equality is not required for coinitial.)
Where Should You Go Next After Coinitial Vectors?
Classifying vectors is the doorway into vector algebra, and a few natural next steps open from here.
Types of vectors. See the full family, unit, zero, position, equal, and negative vectors, so every classification sits in one map instead of being learned one label at a time.
Addition of vectors. Put the coinitial setup to work with the triangle and parallelogram laws, the exact place a shared start becomes essential.
Components of a vector. Break any vector into its horizontal and vertical parts, the skill that turns arrows into numbers you can compute with.
If your child is building these foundations, a live Bhanzu trainer teaches vectors starting from the picture (arrows, tails, and shared starting points) before any formula, in the Bhanzu algebra program.
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