Question

If $|\vec a|=5$ and $-2\le \lambda \le 1$, then the sum of the greatest and the smallest value of $|\lambda \vec a|$ is

• A. $-5$
• B. $5$
• C. $10$
• D. $15$

Hint:

For vectors multiplied by scalars, always use \(|k\vec a|=|k||\vec a|\). Then determine maximum and minimum values using the given interval.

Answer 1

The Correct Option is C

To solve the problem of finding the sum of the greatest and the smallest values of \(|\lambda \vec{a}|\), we need to first understand how scalar multiplication affects the magnitude of a vector. Given that \(|\vec{a}| = 5\) and \(-2 \leq \lambda \leq 1\), we must examine the expression \(|\lambda \vec{a}| = |\lambda| |\vec{a}|\).

1. Calculate \(|\lambda| |\vec{a}|\):
   • The magnitude of a vector scaled by a scalar \(\lambda\) is the absolute value of the scalar times the magnitude of the vector.
   • Thus, \(|\lambda \vec{a}| = |\lambda| \cdot 5\).
2. Determine the range of \(|\lambda|:\)
   • The given range for \(\lambda\) is \(-2 \leq \lambda \leq 1\).
   • Therefore, the range for \(|\lambda|\) would be \(0 \leq |\lambda| \leq 2\) (since the absolute value of a number is always non-negative and maximum covers both ends).
3. Find the smallest and greatest values of \(|\lambda| |\vec{a}|:\)
   • Smallest value: When \(|\lambda| = 0\), \(|\lambda \vec{a}| = 0 \times 5 = 0\).
   • Greatest value: When \(|\lambda| = 2\), \(|\lambda \vec{a}| = 2 \times 5 = 10\).
4. Calculate the sum of the smallest and greatest values:
   • Sum = 0 + 10 = 10.

Therefore, the sum of the greatest and the smallest value of \(|\lambda \vec{a}|\) is \(10\). The correct answer is 10.