The problem asks you to determine the magnitude of the projected component of the force vector F along cable AB using the dot product approach.

Given:

• Force vector: $\mathbf{F} = 64\mathbf{i} + 11\mathbf{j} - 43\mathbf{k}$ N
• Unit vector along cable AB: $\mathbf{u}_{AB} = -0.9231\mathbf{i} - 0.2308\mathbf{j} + 0.3077\mathbf{k}$

Solution approach:

The component of the force F along the direction of AB can be calculated using the dot product formula: $F_{AB} = \mathbf{F} \cdot \mathbf{u}_{AB}$ Where: $F_{AB} = (64)(-0.9231) + (11)(-0.2308) + (-43)(0.3077)$

Now, let's compute:

1. $64 \times (-0.9231) = -59.0784$
2. $11 \times (-0.2308) = -2.5388$
3. $-43 \times 0.3077 = -13.2251$

Summing these values gives: $F_{AB} = -59.0784 - 2.5388 - 13.2251 = -74.8423 \, \text{N}$

The magnitude of the projected component is $\boxed{74.84 \, \text{N}}$.

Do you need further details or explanations?

Here are 5 related questions to deepen your understanding:

1. How does the dot product help in finding the component of a vector along a direction?
2. Why is the unit vector used in this calculation?
3. What is the significance of the negative sign in the result?
4. How would the calculation change if the force vector had different values?
5. What are some other applications of the dot product in physics and engineering?

Tip:

Always verify that the vector you're projecting onto is a unit vector. If it isn't, you need to normalize it first.