If $\vec{a} imes\vec{b}=\vec{c} imes\vec{d}$ and $\vec{a} imes\vec{c}=\vec{b} imes\vec{d}$, then vectors $\vec{a}-\vec{d}$ and $\vec{b}-\vec{c}$ are

Q: If $\vec{a}\times\vec{b}=\vec{c}\times\vec{d}$ and $\vec{a}\times\vec{c}=\vec{b}\times\vec{d}$, then vectors $\vec{a}-\vec{d}$ and $\vec{b}-\vec{c}$ are

If $\vec{a}\times\vec{b}=\vec{c}\times\vec{d}$ and $\vec{a}\times\vec{c}=\vec{b}\times\vec{d}$, then vectors $\vec{a}-\vec{d}$ and $\vec{b}-\vec{c}$ are

Question

If $\vec{a}	imes\vec{b}=\vec{c}	imes\vec{d}$ and $\vec{a}	imes\vec{c}=\vec{b}	imes\vec{d}$, then vectors $\vec{a}-\vec{d}$ and $\vec{b}-\vec{c}$ are

Accepted Answer

To determine why vectors $\vec{a} - \vec{d}$ and $\vec{b} - \vec{c}$ are perpendicular, we need to analyze the given vector equations and their implications.

### Explanation:

1. **Given Equations:**
   - $\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$
   - $\vec{a} \times \vec{c} = \vec{b} \times \vec{d}$

2. **Cross Product Properties:**
   - The cross product of two vectors results in a vector that is perpendicular to both of the original vectors.
   - If two cross products are equal, it implies that the resulting vectors are parallel and have the same magnitude.

3. **Analyzing the Equations:**
   - From $\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$, we know that the vector resulting from $\vec{a} \times \vec{b}$ is parallel to the vector resulting from $\vec{c} \times \vec{d}$.
   - Similarly, from $\vec{a} \times \vec{c} = \vec{b} \times \vec{d}$, the vector resulting from $\vec{a} \times \vec{c}$ is parallel to the vector resulting from $\vec{b} \times \vec{d}$.

4. **Implication for $\vec{a} - \vec{d}$ and $\vec{b} - \vec{c}$:**
   - Consider the vector $\vec{a} - \vec{d}$. This vector can be thought of as a resultant vector from subtracting $\vec{d}$ from $\vec{a}$.
   - Similarly, $\vec{b} - \vec{c}$ is the resultant vector from subtracting $\vec{c}$ from $\vec{b}$.
   - The conditions given by the cross products imply a specific geometric configuration where these resultant vectors are perpendicular to each other.

5. **Why Option 3 is Correct:**
   - The conditions provided by the cross product equations suggest a scenario where the vectors $\vec{a} - \vec{d}$ and $\vec{b} - \vec{c}$ are orthogonal. This is because the cross product conditions imply a perpendicular relationship between the resultant vectors when considering the vector subtraction.

### Why Other Options are Incorrect:

1. **Option 1 (Equal):**
   - There is no information or condition in the given equations that suggests $\vec{a} - \vec{d}$ and $\vec{b} - \vec{c}$ have the same magnitude or direction, which would be necessary for them to be equal.

2. **Option 2 (Parallel):**
   - The cross product conditions imply perpendicularity rather than parallelism. If the vectors were parallel, their cross product would be zero, which contradicts the given conditions.

3. **Option 4 (Inclined at an angle $45^\circ$):**
   - There is no specific information or condition in the given equations that suggests the vectors are inclined at $45^\circ$. The conditions strongly imply perpendicularity, which is a $90^\circ$ angle, not $45^\circ$.

In conclusion, based on the given vector equations and the properties of cross products, the vectors $\vec{a} - \vec{d}$ and $\vec{b} - \vec{c}$ are indeed perpendicular, supporting Option 3 as the correct answer.