If a, b and c are non-zero vectors such that `a times b=c, b times c=a " and "c times a=b` then

Options

A. [a b c] = 0
B. a = b = c
C. `abs(a)=abs(b)=abs(c)`
D. `abs(a)+abs(b)-abs(c)=0`

Answer

To solve the problem, we need to analyze the given vector equations:

1. \( \mathbf{a} \times \mathbf{b} = \mathbf{c} \)
2. \( \mathbf{b} \times \mathbf{c} = \mathbf{a} \)
3. \( \mathbf{c} \times \mathbf{a} = \mathbf{b} \)

We want to find the relationship between the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \).

### Step 1: Take the dot product of the first equation with \( \mathbf{c} \)

From the first equation, we have:
\[ \mathbf{a} \times \mathbf{b} = \mathbf{c} \]
Taking the dot product with \( \mathbf{c} \):
\[ (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{c} \]
Using the property of the scalar triple product, we know that:
\[ (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
Thus, we can rewrite the equation as:
\[ \mathbf{a} \cdot \mathbf{a} = |\mathbf{c}|^2 \]

### Step 2: Take the dot product of the second equation with \( \mathbf{a} \)

From the second equation, we have:
\[ \mathbf{b} \times \mathbf{c} = \mathbf{a} \]
Taking the dot product with \( \mathbf{a} \):
\[ (\mathbf{b} \times \mathbf{c}) \cdot \mathbf{a} = \mathbf{a} \cdot \mathbf{a} \]
Using the scalar triple product property again:
\[ (\mathbf{b} \times \mathbf{c}) \cdot \mathbf{a} = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) \]
Thus, we can rewrite the equation as:
\[ |\mathbf{a}|^2 = |\mathbf{b}|^2 \]

### Step 3: Take the dot product of the third equation with \( \mathbf{b} \)

From the third equation, we have:
\[ \mathbf{c} \times \mathbf{a} = \mathbf{b} \]
Taking the dot product with \( \mathbf{b} \):
\[ (\mathbf{c} \times \mathbf{a}) \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{b} \]
Using the scalar triple product property:
\[ (\mathbf{c} \times \mathbf{a}) \cdot \mathbf{b} = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) \]
Thus, we can rewrite the equation as:
\[ |\mathbf{b}|^2 = |\mathbf{c}|^2 \]

### Step 4: Combine the results

From the above steps, we have established the following relationships:
1. \( |\mathbf{a}|^2 = |\mathbf{c}|^2 \)
2. \( |\mathbf{b}|^2 = |\mathbf{a}|^2 \)
3. \( |\mathbf{b}|^2 = |\mathbf{c}|^2 \)

This implies:
\[ |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| \]

### Conclusion

Thus, the magnitudes of the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \) are equal, but the vectors themselves are not necessarily equal.