Consider a vector \(\vec{v}\) defined in a 2D coordinate system as \(\vec{v} = (3, 4)\).
(a) Express \(\vec{v}\) in polar coordinates and calculate its magnitude and direction.
(b) Now, extend this vector to a 3D coordinate system by assigning a \(z\)-component of 5. Write the new vector \(\vec{v_3}\) in component form, and determine its magnitude.
(c) Discuss how the vector transformation from 2D to 3D changes the interpretation of its components and what challenges arise in mixed reference frames.
Question
Express \(\vec{v} = (3, 4)\) in polar coordinates by finding its magnitude and direction (angle measured from the positive x-axis).
Hint
Hint
Hint
Recall that \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(y/x)\).
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Complete Answer
Magnitude: 5; Direction: \(\theta = \tan^{-1}(\frac{4}{3})\) (approximately \(53.13^\circ\))
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Question
Extend the 2D vector \(\vec{v}\) to 3D by adding a \(z\)-component of 5, forming \(\vec{v_3} = (3, 4, 5)\). Calculate its magnitude.
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Hint
Use the 3D magnitude formula: \(\sqrt{x^2 + y^2 + z^2}\).
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Complete Answer
Magnitude: \(\sqrt{3^2 + 4^2 + 5^2} = \sqrt{9+16+25} = \sqrt{50} \approx 7.07\)
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Question
Discuss the impact of extending a 2D vector into 3D on the interpretation of its components and the challenges that may arise when working in mixed reference frames.
Hint
Hint
Hint
Consider the additional complexity introduced by the third dimension and issues in coordinate transformation.
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- Formula Booklet
Complete Answer
Extending a 2D vector to 3D adds an extra component that complicates the analysis of its direction and magnitude. This introduces challenges in ensuring consistent reference frames and managing rotations or transformations between coordinate systems.
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