Let  ,  and  be three given vectors. If  is a vector such that  and  then  is equal to

To find the vector $\mathbf{c}$ given the conditions $\mathbf{a} + \mathbf{b} = \mathbf{c}$ and $\mathbf{b} + \mathbf{c} = \mathbf{a}$, we can follow the steps below:

1. Break Down the Problem

We are given two equations involving vectors:

1. $\mathbf{a} + \mathbf{b} = \mathbf{c}$
2. $\mathbf{b} + \mathbf{c} = \mathbf{a}$

2. Relevant Concepts

To solve for $\mathbf{c}$, we need to express it in terms of $\mathbf{a}$ and $\mathbf{b}$ from the first equation and substitute it into the second equation.

3. Analysis and Detail

Substituting $\mathbf{c}$ from the first equation into the second: $\mathbf{c} = \mathbf{a} + \mathbf{b}$

Substitute this into the second equation: $\mathbf{b} + (\mathbf{a} + \mathbf{b}) = \mathbf{a}$

This simplifies to: $\mathbf{b} + \mathbf{a} + \mathbf{b} = \mathbf{a}$ $2\mathbf{b} + \mathbf{a} = \mathbf{a}$

From here we can isolate $\mathbf{b}$: $2\mathbf{b} = \mathbf{0} \implies \mathbf{b} = \mathbf{0}$

4. Verify and Summarize

Plugging $\mathbf{b} = \mathbf{0}$ back into the original equations:

• From $\mathbf{a} + \mathbf{0} = \mathbf{c}$, we find $\mathbf{c} = \mathbf{a}$.
• From $\mathbf{0} + \mathbf{c} = \mathbf{a}$, it confirms that $\mathbf{c} = \mathbf{a}$.

Final Answer

Thus, the vector $\mathbf{c}$ is equal to $\mathbf{a}$: $\mathbf{c} = \mathbf{a}$