Question:
If \( |\vec{a}| = 5 \) and \( -2 \leq \lambda \leq 1 \), then the sum of greatest and the smallest value of \( |\lambda \vec{a}| \) is
If \( |\vec{a}| = 5 \) and \( -2 \leq \lambda \leq 1 \), then the sum of greatest and the smallest value of \( |\lambda \vec{a}| \) is
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Magnitude is always a non-negative quantity.
When dealing with intervals for absolute values, check if zero lies within the interval; if it does, the minimum absolute value is 0.
When dealing with intervals for absolute values, check if zero lies within the interval; if it does, the minimum absolute value is 0.
Updated On: Sep 10, 2026
- \( -5 \)
- \( 5 \)
- \( 10 \)
- \( 15 \)
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The Correct Option is C
Solution and Explanation
Concept:
• The magnitude of a scalar multiple of a vector is \( |k\vec{a}| = |k| \cdot |\vec{a}| \).
• Absolute value \( |\lambda| \) represents the distance from zero on the number line.
Step 1: Express the magnitude in terms of \( \lambda \) and \( |\vec{a}| \)
Given \( |\vec{a}| = 5 \).
The expression to evaluate is \( f(\lambda) = |\lambda \vec{a}| = |\lambda| \cdot |\vec{a}| = 5|\lambda| \).
Step 2: Determine the range of \( |\lambda| \)
Given the interval \( -2 \leq \lambda \leq 1 \).
The absolute value \( |\lambda| \) will range from its minimum value at zero to its maximum value at the endpoint furthest from zero.
Minimum \( |\lambda| = 0 \) (at \( \lambda = 0 \)).
Maximum \( |\lambda| = |-2| = 2 \).
Step 3: Find the greatest and smallest values of \( 5|\lambda| \)
Smallest value \( = 5 \times 0 = 0 \).
Greatest value \( = 5 \times 2 = 10 \).
Step 4: Calculate the sum
Sum \( = \text{Greatest Value} + \text{Smallest Value} \)
Sum \( = 10 + 0 = 10 \).
This matches option (C).
• The magnitude of a scalar multiple of a vector is \( |k\vec{a}| = |k| \cdot |\vec{a}| \).
• Absolute value \( |\lambda| \) represents the distance from zero on the number line.
Step 1: Express the magnitude in terms of \( \lambda \) and \( |\vec{a}| \)
Given \( |\vec{a}| = 5 \).
The expression to evaluate is \( f(\lambda) = |\lambda \vec{a}| = |\lambda| \cdot |\vec{a}| = 5|\lambda| \).
Step 2: Determine the range of \( |\lambda| \)
Given the interval \( -2 \leq \lambda \leq 1 \).
The absolute value \( |\lambda| \) will range from its minimum value at zero to its maximum value at the endpoint furthest from zero.
Minimum \( |\lambda| = 0 \) (at \( \lambda = 0 \)).
Maximum \( |\lambda| = |-2| = 2 \).
Step 3: Find the greatest and smallest values of \( 5|\lambda| \)
Smallest value \( = 5 \times 0 = 0 \).
Greatest value \( = 5 \times 2 = 10 \).
Step 4: Calculate the sum
Sum \( = \text{Greatest Value} + \text{Smallest Value} \)
Sum \( = 10 + 0 = 10 \).
This matches option (C).
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