If `a+b+c=0, "then "axxb ` is equal to

Options

A. `bxxc`
B. `cxxb`
C. `axxc`
D. None of these

Explanation

To solve the problem, we start with the given equation involving vectors:

**Given:**
\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = 0 \]

From this equation, we can express one vector in terms of the others. Let's isolate \(\mathbf{c}\):

\[ \mathbf{c} = -(\mathbf{a} + \mathbf{b}) \]

Now, we need to find the value of \(\mathbf{a} \times \mathbf{b}\). To do this, we will take the cross product of both sides of the equation \(\mathbf{c} = -(\mathbf{a} + \mathbf{b})\) with \(\mathbf{a}\):

\[ \mathbf{c} \times \mathbf{a} = -(\mathbf{a} + \mathbf{b}) \times \mathbf{a} \]

Using the distributive property of the cross product, we have:

\[ \mathbf{c} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{a} + \mathbf{b} \times \mathbf{a}) \]

Since the cross product of any vector with itself is zero, \(\mathbf{a} \times \mathbf{a} = 0\):

\[ \mathbf{c} \times \mathbf{a} = -\mathbf{b} \times \mathbf{a} \]

Now, we can rearrange this equation to express \(\mathbf{a} \times \mathbf{b}\):

\[ \mathbf{b} \times \mathbf{a} = -\mathbf{c} \times \mathbf{a} \]

Next, we can take the cross product of both sides of the original equation \(\mathbf{c} = -(\mathbf{a} + \mathbf{b})\) with \(\mathbf{b}\):

\[ \mathbf{c} \times \mathbf{b} = -(\mathbf{a} + \mathbf{b}) \times \mathbf{b} \]

Again, applying the distributive property:

\[ \mathbf{c} \times \mathbf{b} = -(\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{b}) \]

Since \(\mathbf{b} \times \mathbf{b} = 0\):

\[ \mathbf{c} \times \mathbf{b} = -\mathbf{a} \times \mathbf{b} \]

Now, we have two equations:
1. \(\mathbf{b} \times \mathbf{a} = -\mathbf{c} \times \mathbf{a}\)
2. \(\mathbf{c} \times \mathbf{b} = -\mathbf{a} \times \mathbf{b}\)

From these equations, we can conclude that:

\[ \mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{b} \]

Thus, we have established that:

\[ \mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a} \]

This means that \(\mathbf{a} \times \mathbf{b}\) is equal to \(\mathbf{b} \times \mathbf{c}\) or \(\mathbf{c} \times \mathbf{a}\).

**Final Result:**
\[ \mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a} \]