If `vec(b) and vec(c )` are the position vectors of the points B and C respectively, then the position vector of the point D such that `vec(BD)= 4 vec(BC)` is

Options

A. `4 (vec(c )- vec(b))`
B. `-4 (vec(c )- vec(b))`
C. `4vec(c ) - 3vec(b)`
D. `4vec(c ) + 3 vec(b)`

Answer

To solve the problem, we need to find the position vector of point D given that \( \vec{BD} = 4 \vec{BC} \).

### Step 1: Understand the given vectors
Let:
- \( \vec{b} \) be the position vector of point B.
- \( \vec{c} \) be the position vector of point C.
- \( \vec{d} \) be the position vector of point D.

### Step 2: Express the vectors in terms of position vectors
We know that:
- The vector \( \vec{BD} \) can be expressed as:
\[ \vec{BD} = \vec{d} - \vec{b} \]

- The vector \( \vec{BC} \) can be expressed as:
\[ \vec{BC} = \vec{c} - \vec{b} \]

### Step 3: Substitute the expressions into the given relation
According to the problem, we have:
\[ \vec{BD} = 4 \vec{BC} \]
Substituting the expressions we derived:
\[ \vec{d} - \vec{b} = 4(\vec{c} - \vec{b}) \]

### Step 4: Expand the equation
Expanding the right-hand side:
\[ \vec{d} - \vec{b} = 4\vec{c} - 4\vec{b} \]

### Step 5: Rearrange the equation
Now, we can rearrange the equation to isolate \( \vec{d} \):
\[ \vec{d} = 4\vec{c} - 4\vec{b} + \vec{b} \]
\[ \vec{d} = 4\vec{c} - 3\vec{b} \]

### Final Result
Thus, the position vector of point D is:
\[ \vec{d} = 4\vec{c} - 3\vec{b} \]