For non-zero vectors `overline(a), overline(b), overline(c), (overline(a)timesoverline(b))*overline(c)=|overline(a)||overline(b)||overline(c)|` holds, iff

Options

A. `overline(a)*overline(b)=0, overline(b)*overline(c)=0, overline(c)*overline(a)ne0`
B. `overline(a)*overline(b)ne0, overline(b)*overline(c)=0, overline(c)*overline(a)=0`
C. `overline(a)*overline(b)=0, overline(b)*overline(c)ne0, overline(c)*overline(a)=0`
D. `overline(a)*overline(b)=0, overline(b)*overline(c)=0, overline(c)*overline(a)=0`

Answer

To solve the problem, we need to determine the condition under which the equation \((\overline{a} \times \overline{b}) \cdot \overline{c} = |\overline{a}| |\overline{b}| |\overline{c}|\) holds true for non-zero vectors \(\overline{a}\), \(\overline{b}\), and \(\overline{c}\).

### Step-by-Step Solution:

1. **Understanding the Cross Product and Dot Product**:
The left-hand side of the equation involves the cross product of vectors \(\overline{a}\) and \(\overline{b}\) followed by the dot product with vector \(\overline{c}\). The cross product \(\overline{a} \times \overline{b}\) results in a vector that is perpendicular to both \(\overline{a}\) and \(\overline{b}\).

2. **Expressing the Left-Hand Side**:
We can express the left-hand side as:
\[ (\overline{a} \times \overline{b}) \cdot \overline{c} = |\overline{a} \times \overline{b}| |\overline{c}| \cos \theta \]
where \(\theta\) is the angle between the vector \(\overline{c}\) and the vector \(\overline{a} \times \overline{b}\).

3. **Magnitude of the Cross Product**:
The magnitude of the cross product \(|\overline{a} \times \overline{b}|\) can be expressed as:
\[ |\overline{a} \times \overline{b}| = |\overline{a}| |\overline{b}| \sin \phi \]
where \(\phi\) is the angle between vectors \(\overline{a}\) and \(\overline{b}\).

4. **Substituting into the Equation**:
Substituting the magnitude of the cross product into the left-hand side gives:
\[ |\overline{a}| |\overline{b}| \sin \phi |\overline{c}| \cos \theta \]

5. **Setting Up the Equation**:
Now we equate this to the right-hand side:
\[ |\overline{a}| |\overline{b}| \sin \phi |\overline{c}| \cos \theta = |\overline{a}| |\overline{b}| |\overline{c}| \]

6. **Canceling Common Terms**:
Since \(\overline{a}\), \(\overline{b}\), and \(\overline{c}\) are non-zero vectors, we can safely divide both sides by \(|\overline{a}| |\overline{b}| |\overline{c}|\):
\[ \sin \phi \cos \theta = 1 \]

7. **Analyzing the Equation**:
The maximum value of \(\sin \phi\) and \(\cos \theta\) is 1. For the product \(\sin \phi \cos \theta\) to equal 1, both \(\sin \phi\) and \(\cos \theta\) must equal 1. This leads to:
- \(\sin \phi = 1\) implies \(\phi = 90^\circ\) (the angle between \(\overline{a}\) and \(\overline{b}\) is 90 degrees).
- \(\cos \theta = 1\) implies \(\theta = 0^\circ\) (the vector \(\overline{c}\) is parallel to \(\overline{a} \times \overline{b}\)).

8. **Conclusion**:
Therefore, the condition under which the equation holds is that the vectors are mutually perpendicular:
- \(\overline{a} \perp \overline{b}\)
- \(\overline{b} \perp \overline{c}\)
- \(\overline{c} \perp \overline{a}\)