GATE EC 2025 Set 1 — Question 59
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Engineering Mathematics → Calculus → Maxima & Minima
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Question
Consider the vectorsFor real-valued scalar variable , the value ofis ____________ (rounded off to two decimal places).
denotes the Euclidean norm, i.e., for .
denotes the Euclidean norm, i.e., for .
Correct answer
2.95 to 3.05
Solution
We want to find the minimum value of the Euclidean norm . This problem can be interpreted as finding the minimum distance from the vector to the subspace (line) spanned by vector .Method 1: Calculus
Given:The vector is:The squared Euclidean norm is:To find the minimum, we take the derivative of with respect to and set it to zero:Substitute back into :The minimum value of the norm is:Method 2: Geometric Interpretation
The vector represents the line in the 2D plane. The vector represents the point . The minimum distance from a point to a line is given by:For the line and point :Rounding to two decimal places, the answer is 3.00.
Given:The vector is:The squared Euclidean norm is:To find the minimum, we take the derivative of with respect to and set it to zero:Substitute back into :The minimum value of the norm is:Method 2: Geometric Interpretation
The vector represents the line in the 2D plane. The vector represents the point . The minimum distance from a point to a line is given by:For the line and point :Rounding to two decimal places, the answer is 3.00.
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