From the image provided, the problem asks to identify the correct statements about vector relationships. Let's analyze each option and clarify the situations where they hold true:

1. $|\vec{c} - \vec{d}| = |\vec{c}| + |\vec{d}|$ then and only then when the vectors $\vec{c}$ and $\vec{d}$ are oppositely directed:

   • This is true because the equation implies the maximum possible subtraction magnitude, which occurs when vectors are oppositely aligned (i.e., angle between them is $180^\circ$).

2. $|\vec{c} + \vec{d}| = |\vec{c}| + |\vec{d}|$ then and only then when the vectors $\vec{c}$ and $\vec{d}$ are co-directed:

   • This is also true because the equation describes the maximum addition magnitude, achievable only when the vectors are parallel and aligned in the same direction ($0^\circ$).

3. For any vectors $\vec{c}$ and $\vec{d}$: (|\vec{c} - \vec{d}| \leq |\vec{c}| + |\vec{d}|]:

   • True by the triangle inequality for vector subtraction.

4. For any vectors $\vec{c}$ and $\vec{d}$: (|\vec{c} + \vec{d}| \leq |\vec{c}| + |\vec{d}|]:

   • True by the triangle inequality for vector addition.

5. $|\vec{c} + \vec{d}| = |\vec{c}| + |\vec{d}|$ then and only then when the vectors $\vec{c}$ and $\vec{d}$ are co-directed:

   • This repeats option 2 and is correct.

Final Correct Options:

• 1
• 2
• 3
• 4

Would you like a detailed mathematical explanation of any specific statement? Let me know!

Related Questions:

1. How does the triangle inequality apply to vectors?
2. Why does $|\vec{c} + \vec{d}|$ reach its maximum when vectors are co-directed?
3. What happens geometrically when vectors are oppositely directed?
4. Can you provide an example of vectors where $|\vec{c} - \vec{d}|$ is strictly less than $|\vec{c}| + |\vec{d}|$?
5. How does the magnitude of a vector sum differ from its components?

Tip:

Always visualize vector addition and subtraction geometrically to understand their magnitude relationships better!