Chapter 2 – Excerpt 3
This short excerpt is just about cross products, including the well-known Right Paw Rule, which humans call the Right Hand Rule.
Actually, the torque vector is the cross product of the radial vector and the force vector. Since cross products are typically studied in 2nd year calculus, you probably are not familiar with them yet. If tensors made you feel tense, cross products may make you feel cross, but don’t worry – they’re much simpler.
There are two ways to multiply vectors – the dot product and the cross product. The operator symbols and x are written in bold to indicate that they are operations on vectors. The dot product yields a scalar, i.e. a number, whereas the cross product yields another vector.
The magnitude of the cross product is the product of the magnitudes of the two vectors and the sine of the angle between them. This English mouthful may be easier to take in as an equation:
The direction of the resultant vector is perpendicular to the plane of the two vectors being crossed. Of course, there are always two perpendicular directions, for example up and down. Which of the two possible directions is given by the canonical Right Paw Rule, as follows:
Note that this rule means that the direction depends on the order of the operands. The cross product is not a commutative operation. In fact, V2 x V1 = – V1 x V2.
Q. What are î x j, j x k and î x k ?
Thus, recalling expression (3) above, we have
τ = r x F
Note that the torque τ is perpendicular to both r and F, and hence it is perpendicular to the entire plane of motion. So the direction of the torque is always along the axis of rotation. Assuming that that axis is vertical, the torque points upward when the induced rotation or revolution is counterclockwise, and downward when it’s clockwise.
Although these quantities are vectors, we can often just use scalars instead, because the rotation can be in one of only two directions – clockwise or counterclockwise – with respect to the axis of rotation. By convention, the counterclockwise direction is considered positive and the clockwise direction negative, as mentioned above.
3 – Polydactyl is the technical term for the colloquial “double-pawed”, which is obviously a misnomer, since you are all most likely quadruple-pawed. Polydactyl cats are also known as “six-finger cats”, “thumb cats”, or “mitten kittens”. <return-to-text>
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