A vector A has components A1, A2, A3 in a right-handed...

A vector $A$ has components $A_{1}, A_{2}, A_{3}$ in a right-handed rectangular cartesian coordinate system $O x, O y, O z$. The coordinate system is rotated about the z-axis through an angle $\frac{\pi}{2} .$ The components of $A$ in the new coordinate system are (A) $A_{1},-A_{2}, A_{3}$ (B) $A_{2}, A_{1}, A_{3}$ (C) $A_{1}, A_{2},-A_{3}$ (D) $A_{2},-A_{1}, A_{3}$.

A vector $A$ has components $A_{1}, A_{2}, A_{3}$ in a right-handed rectangular cartesian coordinate system $O x, O y, O z$. The coordinate system is rotated about the z-axis through an angle $\frac{\pi}{2} .$ The components of $A$ in the new coordinate system are
(A) $A_{1},-A_{2}, A_{3}$
(B) $A_{2}, A_{1}, A_{3}$
(C) $A_{1}, A_{2},-A_{3}$
(D) $A_{2},-A_{1}, A_{3}$.

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Key Concepts

Cartesian Coordinate Systems

A Cartesian coordinate system consists of mutually perpendicular axes that define the position of points in space using coordinate values. It provides a framework where vectors can be expressed as components along the x, y, and z axes. Understanding this system is fundamental to analyzing vector quantities and their transformations under rotations or reflections.

Rotation Matrices

Rotation matrices are used to transform the coordinates of vectors when the coordinate system is rotated. They are square matrices whose elements are determined by the angle of rotation, and they preserve the lengths and angles between vectors. In two dimensions, a rotation matrix for an angle ? is typically given by [[cos?, sin?], [-sin?, cos?]]. This matrix is key in converting vector components from one coordinate system to another via rotation.

Vector Transformation under Rotation

When a vector is expressed in one coordinate system and the system is rotated, the vector's components must be transformed according to the rotation matrix associated with that rotation. This process reassigns the original components to their new directions in the rotated system, ensuring that the physical vector remains unchanged even though its components relative to the new axes may look different.

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