Question

A situation may be described by using different sets of coordinate axes having different orientations. Magnitude of which of the following do not depend on the orientation of the axis ?

• A. Dot product
• B. Cross product
• C. Work done by force
• D. Torque by a force

Answer 1

Answer: (A), (B), (C), (D)

Torque by a force

Let us consider two vectors as,
$\overrightarrow{A}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}$

$\overrightarrow{B}=x_2\hat{i}+y_2\hat{j}+z_2\hat{k}$

We know that the dot product, $\overrightarrow{A}.\overrightarrow{B}=x_1{x}_2+y_1{y}_2+z_1{z}_2=AB\cos \theta ...\left(1\right)$

Cross product,

$\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ x_1& y_1& z_1\\ x_2& y_2& z_2\end{array}\right|=AB\sin \theta ...\left(2\right)$

Also, for a given vector, it's magnitude and direction is independent of orientation of the axes.

Thus, angle between two vectors is also independent of orientation of the axes.
Based on these observations and from $\left(1\right)$ and $\left(2\right)$ dot product and cross product are independent of orientation of the axis.

Hence, all the options are correct.

Further,
Work done by a force is dot product of force and displacement
$w=\overrightarrow{F}.\overrightarrow{d}$

Torque by a force is cross product of force and position vector from reference point.
$\tau =\overrightarrow{r}\times \overrightarrow{F}$
Hence, work done and torque are also independent of orientation of the axes.