The problem asks to calculate the moment vector $M$ of force $p$ about point $Q$, given that the force $p$ acts through point $A$. The points $p$, $A$, and $Q$ are provided in two different cases.

The moment vector $M$ is calculated using the cross product formula:

$M = r \times p$

Where:

• $r$ is the position vector from point $Q$ to point $A$, i.e., $r = \vec{QA} = A - Q$,
• $p$ is the force vector.

Case (a):

• Force $p = [0, 5, 0]$,
• $A = (2, 3, 0)$,
• $Q = (0, 0, 0)$.

In this case, $r = A - Q = (2, 3, 0)$.

Now, calculate $M$ using the cross product $M = r \times p$.

Case (b):

• Force $p = [8, 6, 4]$,
• $A = (2, 2, 3)$,
• $Q = (10, 8, 7)$.

Here, $r = A - Q = (2, 2, 3) - (10, 8, 7) = (-8, -6, -4)$.

Now, calculate $M = r \times p$ for this case.

I will calculate these moments next.The moment vectors for the two cases are:

Case (a):

• The moment vector $M$ is $[0, 0, 10]$.

Case (b):

• The moment vector $M$ is $[0, 0, 0]$, meaning there is no moment in this case.

Would you like further details or explanations on the calculations? Here are some related questions you might explore:

1. How is the cross product used to calculate the moment of a force?
2. Why does the moment in Case (b) result in zero?
3. What is the physical interpretation of the moment vector?
4. How can the direction of the moment vector be interpreted in 3D space?
5. What is the difference between torque and moment?

Tip: The magnitude of the moment (or torque) depends on both the magnitude of the force and the perpendicular distance from the pivot point.