Final answer:
To express vector A = ar (3cosφ) - aφ (2r) + az (5) in Cartesian coordinates, use the conversion formulas x = r cosφ, y = r sinφ, z = z. The Cartesian vector is A = (ar (3cosφ) - aφ (2r)sinφ)î + (aφ (2r)cosφ)ˇ + (az (5))k.
Step-by-step explanation:
The student has asked how to express a vector A = ar (3cosφ) - aφ (2r) + az (5) in Cartesian coordinates x, y, z. This is typically done by converting from cylindrical coordinates (r, φ, z) to Cartesian coordinates (x, y, z). The conversion formulas are x = r cosφ, y = r sinφ, and z = z. Using these, we can rewrite each component of the vector A. For aR, the x-component becomes Ax = ar (3cosφ) and the y-component is Ay = 0 since there is no sine term. For aφ, the x-component is Ax = -aφ (2r)sinφ, and the y-component is Ay = aφ (2r)cosφ. Lastly, for az, the z-component is simply Az = az (5). Therefore, the vector A in Cartesian coordinates is the sum A = (ar (3cosφ) - aφ (2r)sinφ)î + (aφ (2r)cosφ)ˇ + (az (5))k.