So you’re staring at two vector equations — vxy and vwz — and someone asks you, “What is vw?That's why this little puzzle trips up a lot of people, especially when they’re first learning to think in vectors. ” You’re not alone. But it’s not. Which means that’s not how this works. ” And you’re thinking, “Wait, what? It’s actually a neat test of whether you really get what those subscripts mean. But it sounds like a trick question, or maybe a typo. So let’s break it down, because once you see it, you can’t unsee it.
What Is This Even Asking?
First, let’s translate the notation. Also, when you see something like vxy, that’s not a single vector. It’s a way of writing the vector from point X to point Y, but with a twist — the “v” is just a label, like velocity or displacement, and the subscripts tell you the start and end points. So vxy means “the vector from X to Y.” Similarly, vwz means “the vector from W to Z Less friction, more output..
No fluff here — just what actually works.
Now the question: given vxy and vwz, what is vw?
At first glance, it looks like we’re trying to combine two unrelated vectors — one from X to Y, another from W to Z — and somehow get a vector from V to W. That doesn’t make sense unless there’s a relationship between these points. And that’s the key. Consider this: the notation assumes you’re working within a connected system — like a path or a chain of points. You can’t just pull vw out of thin air from those two pieces of information unless you know how X, Y, W, and Z relate to V and W It's one of those things that adds up. Still holds up..
The Subscript Language
Think of each vector as a directed line segment. So vxy goes from X to Y. Day to day, the first subscript is the starting point, the second is the ending point. If you want to get from V to W, you need a direct connection — either you have that vector given, or you can build it by adding other vectors that connect V to W through intermediate points.
Why This Confuses People
Here’s where most folks get stuck: they treat the letters as abstract symbols instead of actual points in space. They see vxy and vwz and think, “Okay, I have two vectors, now what?” But the question isn’t about vector arithmetic yet — it’s about reading the map But it adds up..
The confusion usually comes from not seeing the bigger picture. Are X, Y, W, Z all part of the same diagram? Is there a path from V to W that goes through X, Y, or Z? Practically speaking, without that context, vw is undefined. You might as well ask, “Given the distance from New York to Chicago and from Los Angeles to Miami, what’s the distance from Boston to Seattle?” There’s no direct relationship unless you connect the cities in some way.
The Missing Link
In many problems, the points are arranged in a chain or a polygon. Here's the thing — for example, maybe you have points V, X, Y, W, Z in a line or a shape. Worth adding: if that’s the case, you can use vector addition to find vw by combining other known vectors. But the problem statement “given vxy and vwz” doesn’t provide that arrangement — it just gives you two isolated pieces.
Not obvious, but once you see it — you'll see it everywhere.
So the real answer to “what is vw?And ” might be: “We can’t determine vw from just vxy and vwz unless we know how these points are connected. ” And that’s a perfectly valid answer. Sometimes the most important insight is recognizing what you don’t know And that's really what it comes down to..
How to Actually Find vw (When You Can)
Let’s assume there is a connection — because in most textbook or exam problems, there is. Usually, the points are arranged so that you can express vw as a combination of the given vectors. Here’s how you think it through:
Step 1: Draw It Out
Sketch the points. Ask: Is there a path from V to W that passes through X, Y, or Z? Don’t worry about scale — just get the relationships clear. Which means label them V, X, Y, W, Z. Maybe V connects to X, X to Y, Y to W, and W to Z The details matter here..
vv (trivial, zero vector)
vx (unknown)
vxy (known)
vyw (maybe unknown)
vwz (known)
But you need to express vw in terms of what you have Worth keeping that in mind..
Step 2: Use Vector Addition Rules
Remember, if you go from point A to B to C, the total vector is vab + vbc = vac. So if you can find a sequence of vectors that add up to vw, you’re golden.
Here's a good example: suppose the points are in order: V, X, Y, W, Z. Then:
vv = 0
vx = ?
vxy is given
vyw = ?
vwz is given
But you want vw. If you can get from V to W via X and Y, then:
vw = vx + vxy + vyw
But you don’t know vx or vyw. So that doesn’t help Less friction, more output..
What if the path is V → Z → W? Then vw = vvz? No, that’s not right.
Maybe the points are arranged differently. On top of that, let’s try a common setup: V, W, X, Y, Z in some network. Consider this: for example, maybe vxy means the vector from X to Y, but you’re supposed to realize that V, X, Y, W are colinear — so vx + xy + yw = vw. Sometimes the problem implies that X, Y, Z are just labels along the way from V to W. If you know vxy (which is vx + xy) and vwz (which might be vw + wz), you still need more That's the part that actually makes a difference..
This is why the problem as stated is ambiguous. Here's the thing — in a real textbook, they’d give you a diagram or say something like “Points V, X, Y, W are collinear” or “Vector vxy represents the displacement from X to Y, and vwz is the displacement from W to Z in a system where V, W, X, Y, Z are vertices of a pentagon. ” Without that, you’re guessing.
Step 3: Look for Algebraic Manipulation
Sometimes you can combine the given vectors algebraically. Suppose you have:
vxy = vy – vx (if you think in position vectors)
vwz = vz – vw
But that introduces position vectors, which might not be what the problem intends. And you still have too many unknowns And that's really what it comes down to..
The truth is, unless there’s additional information — like a geometric relationship, a
geometric constraint, or specific values — you can't uniquely determine vw. That's the honest answer.
Step 4: Make Reasonable Assumptions
In educational settings, problems are designed to be solvable. So let's assume the most common scenario: the points form a connected path where each vector shares an endpoint with the next. This creates a chain: V to X, X to Y, Y to W, W to Z.
Given this arrangement:
- vxy represents the vector from X to Y
- vwz represents the vector from W to Z
- We want vw, the vector from V to W
If we also assume we're given vx (vector from V to X) and vyw (vector from Y to W), then: vw = vx + vxy + vyw
But since we typically don't have all these vectors, we look for what's actually provided. That said, often, problems will give you position vectors instead. If you know the position vectors r⃗₀, r⃗₁, r⃗₂, etc.
We're talking about the bit that actually matters in practice The details matter here..
Step 5: Check Your Work
Always verify that your answer makes sense dimensionally and geometrically. On top of that, does the magnitude seem reasonable? Does the direction align with your sketch? If you can calculate vw two different ways and get the same result, you've likely found the correct approach But it adds up..
Conclusion
Vector problems test both your mathematical skills and your ability to interpret geometric relationships. The key is recognizing that vectors aren't just abstract symbols—they represent real displacements between points in space. When faced with seemingly insufficient information, look for hidden connections: shared endpoints, collinear points, or geometric symmetries that allow you to express unknown vectors in terms of known ones.
Remember, the goal isn't just to find an answer—it's to understand the logical path that leads there. Most vector problems follow a pattern: identify what you know, determine how those elements connect, apply vector addition principles, and verify your solution makes physical sense. With practice, you'll develop an intuition for spotting these connections quickly, turning ambiguous problems into clear, solvable chains of reasoning Worth knowing..
