Derive the expression for torque in cartesian coordinates.
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Torque is a measure of the rotational force applied to an object, and it is given by the product of the force and the perpendicular distance from the axis of rotation to the line of action of the force. In Cartesian coordinates, the torque can be expressed as:
τ = (Fy * z - Fz * y) i + (Fz * x - Fx * z) j + (Fx * y - Fy * x) k
where τ is the torque vector, F is the force vector, and i, j, and k are the unit vectors in the x, y, and z directions, respectively.
The expression can be derived by applying the cross product between the force vector and the position vector (which is the vector from the axis of rotation to the point of application of the force), and then expressing the resulting vector in Cartesian coordinates.
This expression for torque in Cartesian coordinates is essential for analyzing the rotational motion of objects in physics and engineering, and it is used in various applications, such as designing machinery and studying the movement of celestial bodies.
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