GATE EC 2016 Set 2 — Question 15
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Engineering Mathematics → Linear Algebra → Vectors & Vector Operations
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Question
Consider the time-varying vector in Cartesian coordinates, where is a constant. When the vector magnitude is at its minimum value, the angle that makes with the axis (in degrees, such that ) is ________
Correct answer
90 to 90
Solution
To find the angle that the vector makes with the -axis when its magnitude is minimized, we analyze the components of the vector:where:
Step 1: Express the squared magnitude of the vector
The squared magnitude of the vector is given by:Using the trigonometric identity , we can rewrite in terms of :Step 2: Minimize the magnitude
To minimize (and thus minimize ), we must maximize the term being subtracted, which is . Since the maximum value of is , the minimum value of occurs when:This condition implies:At this minimum point, the components of the vector are:Thus, the vector at its minimum magnitude is:Step 3: Calculate the angle with the -axis
The angle that the vector makes with the positive -axis is given by:Since and , the vector lies entirely along the -axis. For :- If , the vector points along the positive -axis, corresponding to .
- If , the vector points along the negative -axis, corresponding to (or ), which lies outside the specified range .
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