Question

Which of the following statements is true?

• A. When the coordinate axes are translated the component of a vector in a plane changes
• B. When the coordinate axes are rotated through some angle components of the vector change but the vector's magnitude remains constant
• C. Sum of $ \vec{a} $ and $ \vec{b} $ is $ \vec{R} $ . If the magnitude of $ \vec{a} $ alone is increased angle between $ \vec{b} $ and $ \vec{R} $ decreases
• D. The cross product of $ 3\,\hat{i} $ and $ 4\,\hat{j} $ is 12

Hint:

Locate the components of a vector in a plane and then move the plane through some angle. Observe the difference in both cases.

Answer 1

The Correct Option is B

Consider a vector A and its components Asin \(\theta\) and \(Acos\theta\) are represented in XY plane of the cartesian coordinate system.

Now when the coordinate system is rotated by an angle \(\alpha\), the new coordinate system \(X'Y'\) is shown in the figure below

From the figure, it is clear that the new component of the vector \(A\) is \(Asin(\theta - \alpha)\) and \(Acos(\theta - \alpha)\) in \(X'Y'\) coordinate plane which is different from the components of the vector \(A\) in \(XY\) coordinate plane.

However in both cases the magnitude of the vector \(A\) is same. i.e. \(|\vec A| = A\)

Therefore when the coordinate axes are rotated through some angle, components of the vector change but the vector's magnitude remains constant.

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