If ` vec a ,\ vec b` are the vectors forming consecutive sides of a regular of a regular hexagon `A B C D E F ,\ ` then the vector representing side `C D` is

Options

A. a) ` vec a+ vec b`
B. b) `vec a- vec b`
C. c) ` vec b- vec a`
D. d) `-( vec a+ vec b)`

Explanation

To find the vector representing side \(CD\) of the regular hexagon \(ABCDEF\) formed by vectors \(\vec{a}\) and \(\vec{b}\), we can follow these steps:

### Step 1: Understand the Geometry of the Hexagon
A regular hexagon has equal sides and symmetrical properties. The vectors \(\vec{a}\) and \(\vec{b}\) represent two consecutive sides of the hexagon.

### Step 2: Identify the Vectors Representing the Sides
Let:
- \(\vec{AB} = \vec{a}\)
- \(\vec{BC} = \vec{b}\)

### Step 3: Use Vector Addition
In a regular hexagon, the vector from point \(A\) to point \(D\) can be expressed as the sum of vectors from \(A\) to \(B\), \(B\) to \(C\), and \(C\) to \(D\):
\[ \vec{AD} = \vec{AB} + \vec{BC} + \vec{CD} \]
This can be rearranged to find \(\vec{CD}\):
\[ \vec{CD} = \vec{AD} - \vec{AB} - \vec{BC} \]

### Step 4: Determine the Vector \(\vec{AD}\)
In a regular hexagon, the vector \(\vec{AD}\) is parallel to \(\vec{BC}\) and has a length that is twice that of \(\vec{BC}\). Therefore, we can express it as:
\[ \vec{AD} = 2\vec{b} \]

### Step 5: Substitute Known Values
Now substitute \(\vec{AD}\), \(\vec{AB}\), and \(\vec{BC}\) into the equation for \(\vec{CD}\):
\[ \vec{CD} = 2\vec{b} - \vec{a} - \vec{b} \]

### Step 6: Simplify the Expression
Combine like terms:
\[ \vec{CD} = 2\vec{b} - \vec{b} - \vec{a} = \vec{b} - \vec{a} \]

### Conclusion
Thus, the vector representing side \(CD\) is:
\[ \vec{CD} = \vec{b} - \vec{a} \]