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

Answer 1

Answer: (B)

When the coordinate axes are rotated through some angle components of the vector change but the vector's magnitude remains constant

Answer 2

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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