Let `vec a` and `vec b` be two vectors,Let `|vec a|=1,|vec b|=4` and `vec a*vec b=2`.If `vec c=(2vec a timesvec b)-3vec b,` then the value of `vec b*vec c` is

Options

A. `-60`
B. `-48`
C. `-84`
D. `-24`

Answer

To solve the problem step by step, we need to find the value of \( \vec{b} \cdot \vec{c} \) where \( \vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b} \).

### Step 1: Understand the given information
We know:
- \( |\vec{a}| = 1 \)
- \( |\vec{b}| = 4 \)
- \( \vec{a} \cdot \vec{b} = 2 \)

### Step 2: Calculate \( \vec{b} \cdot \vec{c} \)
Using the expression for \( \vec{c} \):
\[ \vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b} \]
We can compute \( \vec{b} \cdot \vec{c} \):
\[ \vec{b} \cdot \vec{c} = \vec{b} \cdot (2(\vec{a} \times \vec{b}) - 3\vec{b}) \]
This can be separated into two parts:
\[ \vec{b} \cdot \vec{c} = 2(\vec{b} \cdot (\vec{a} \times \vec{b})) - 3(\vec{b} \cdot \vec{b}) \]

### Step 3: Evaluate \( \vec{b} \cdot (\vec{a} \times \vec{b}) \)
The dot product of a vector with a cross product involving itself is always zero:
\[ \vec{b} \cdot (\vec{a} \times \vec{b}) = 0 \]
Thus, we have:
\[ \vec{b} \cdot \vec{c} = 2(0) - 3(\vec{b} \cdot \vec{b}) = -3(\vec{b} \cdot \vec{b}) \]

### Step 4: Calculate \( \vec{b} \cdot \vec{b} \)
The dot product \( \vec{b} \cdot \vec{b} \) is equal to the square of the magnitude of \( \vec{b} \):
\[ \vec{b} \cdot \vec{b} = |\vec{b}|^2 = 4^2 = 16 \]

### Step 5: Substitute back into the equation
Now substituting back:
\[ \vec{b} \cdot \vec{c} = -3(16) = -48 \]

### Final Answer
Thus, the value of \( \vec{b} \cdot \vec{c} \) is:
\[ \boxed{-48} \]