In vector, why a + b = c in vector triangle?

Question

New to vector and while currently learning the basics my book says according to vector triangle theory a + b = c.But according to geometry in a triangle the sum of two sides are always bigger than the other one.Which means a + b != c.So why OB = a + b? why it does not make any sense to me?

Please explain with simple example.

Answer 1

Yes, in a vector triangle, $\vec{a}+\vec{b}=\vec{c}$.

You seem to have confounded this with the Triangle Inequality.

The triangle inequality, in terms of vectors, states that $|\vec{a}|+|\vec{b}|>|\vec{c}|$, where $|v|$ denotes the magnitude, or length of the vector. The vectors themselves are added, but their magnitudes generally don't.

Here is an example: consider $(3,4)+(2,8)=(5,12)$, but $5+\sqrt{68}>13$.

Answer 2

A vector $\vec{AB}$ is not the same thing as the line $AB$, and neither it is the same thing as the length of the line $AB$.

Perhaps the best way of imagining the vector $\vec{AB}$ is that it is a translation of the plane such that point $A$ is translated/mapped to $B$. Adding vectors is the operation of applying one translation first, then another. Let's say you have three points $A, B, C$. Vector $\vec{AB}$ is the translation that maps the point $A$ into $B$. Vector $\vec{BC}$ is the translation that maps the point $B$ into $C$. When you apply the former, and then the latter, you get a new translation, which happens to map $A$ to $C$, so we call it $\vec{AC}$. That is why:

$$\vec{AB}+\vec{BC}=\vec{AC}$$

For every vector $\vec{AB}$ you can look at all pairs of points $X, Y$ so that $X$ maps into $Y$. There are many of those, and $\vec{AB}=\vec{XY}$ for all of those pairs (i.e. the vector doesn't uniquely determine its "starting" and "ending" point). Still, all those pairs have some common properties: lines $AB$ and $XY$ are:

• Of equal length,
• Parallel to each other (or lie on the same straight line), and
• Of the same, rather than opposite, "direction".

The length of the line $AB$ (or every other such line $XY$) is called the magnitude of the vector $\vec{AB}$ and is denoted $|\vec{AB}|$. In other words, the magnitude of the vector $\vec{AB}$ is $|\vec{AB}|=\text{length}(AB)$. Obviously for three points $A, B, C$ that are not collinear (not on the same straight line), you have:

$$|\vec{AB}|+|\vec{BC}|>|\vec{AC}|$$

This (triangle) inequality is about lengths, and is not in any contradiction with the above equality $\vec{AB}+\vec{BC}=\vec{AC}$, which is about vectors themselves.