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)+3vec(b)`

Explanation

To find the position vector of point D given that \(\vec{BD} = 4 \vec{BC}\), we can follow these steps:

### Step 1: Define the Position 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 \(\vec{BC}\)
The vector \(\vec{BC}\) can be expressed in terms of the position vectors of B and C:
\[ \vec{BC} = \vec{C} - \vec{B} \]

### Step 3: Express \(\vec{BD}\)
According to the problem, we have:
\[ \vec{BD} = 4 \vec{BC} \]
Substituting the expression for \(\vec{BC}\):
\[ \vec{BD} = 4(\vec{C} - \vec{B}) \]

### Step 4: Express \(\vec{BD}\) in Terms of \(\vec{B}\) and \(\vec{D}\)
The vector \(\vec{BD}\) can also be expressed as:
\[ \vec{BD} = \vec{D} - \vec{B} \]

### Step 5: Set the Two Expressions for \(\vec{BD}\) Equal
Now we can set the two expressions for \(\vec{BD}\) equal to each other:
\[ \vec{D} - \vec{B} = 4(\vec{C} - \vec{B}) \]

### Step 6: Rearrange to Solve for \(\vec{D}\)
Rearranging the equation gives:
\[ \vec{D} = \vec{B} + 4(\vec{C} - \vec{B}) \]
Expanding this:
\[ \vec{D} = \vec{B} + 4\vec{C} - 4\vec{B} \]
Combining like terms:
\[ \vec{D} = -3\vec{B} + 4\vec{C} \]

### Final Expression for \(\vec{D}\)
Thus, the position vector of point D is:
\[ \vec{D} = 4\vec{C} - 3\vec{B} \]