If `|vec(a)|=4` and `-3 lek le2` , then the range of `| k vec(a) |` is

Options

A. [10,8]
B. [0,12]
C. [-12,8]
D. [8,12]

Answer

To solve the problem, we need to find the range of the expression \(|k \vec{a}|\) given that \(|\vec{a}| = 4\) and \(-3 \leq k \leq 2\).

### Step-by-Step Solution:

1. **Identify the Magnitude of \(\vec{a}\)**:
We know that \(|\vec{a}| = 4\).

2. **Express \(|k \vec{a}|\)**:
The magnitude of the vector \(k \vec{a}\) can be expressed as:
\[ |k \vec{a}| = |k| \cdot |\vec{a}| \]
Substituting the known value of \(|\vec{a}|\):
\[ |k \vec{a}| = |k| \cdot 4 \]

3. **Determine the Range of \(k\)**:
We are given that \(k\) lies in the range:
\[ -3 \leq k \leq 2 \]

4. **Find the Range of \(|k|\)**:
The absolute value of \(k\) will vary based on the limits of \(k\):
- The minimum value of \(|k|\) occurs at \(k = 0\), which gives \(|k| = 0\).
- The maximum value of \(|k|\) occurs at \(k = -3\) or \(k = 2\), which gives \(|k| = 3\) and \(|k| = 2\) respectively. Therefore, the maximum value of \(|k|\) is \(3\).

5. **Calculate the Range of \(|k \vec{a}|\)**:
Now we can find the range of \(|k \vec{a}|\):
\[ |k \vec{a}| = 4 |k| \]
- The minimum value of \(|k \vec{a}|\) occurs when \(|k| = 0\):
\[ \text{Minimum} = 4 \cdot 0 = 0 \]
- The maximum value of \(|k \vec{a}|\) occurs when \(|k| = 3\):
\[ \text{Maximum} = 4 \cdot 3 = 12 \]

6. **Final Result**:
Thus, the range of \(|k \vec{a}|\) is:
\[ 0 \leq |k \vec{a}| \leq 12 \]
In interval notation, this is:
\[ [0, 12] \]