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

Step 1: Use the property of vector magnitude.
The magnitude of a scalar multiplied vector follows the rule \[ |\lambda \vec a|=|\lambda|\,|\vec a| \] Given \[ |\vec a|=5 \] Thus \[ |\lambda \vec a|=5|\lambda| \] Step 2: Determine the possible values of $\lambda$.
The value of \(\lambda\) lies in the interval \[ -2\le\lambda\le1 \] We must determine the minimum and maximum value of \[ |\lambda| \] Step 3: Find smallest value.
The smallest value of \(|\lambda|\) occurs when \[ \lambda=0 \] Thus \[ |\lambda|=0 \] Therefore \[ |\lambda \vec a|=5\times0=0 \] Step 4: Find largest value.
The largest value of \(|\lambda|\) occurs when \[ \lambda=-2 \] Thus \[ |\lambda|=2 \] Therefore \[ |\lambda \vec a|=5\times2=10 \] Step 5: Find the required sum.
Smallest value \[ 0 \] Largest value \[ 10 \] Sum \[ 0+10=10 \] Step 6: Conclusion.
Thus the sum of the greatest and smallest value of \( |\lambda \vec a| \) is \[ 10 \]