7.4 Q-7

Question Statement

Prove that if:

$$\underline{a} + \underline{b} + \underline{c} = 0$$

then:

$$\underline{\mathbf{a}} \times \underline{\mathbf{b}} = \underline{\mathbf{b}} \times \underline{\mathbf{c}} = \underline{\mathbf{c}} \times \underline{\mathbf{a}}$$

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Background and Explanation

This problem involves vector cross products and properties of vector addition. We will use the fact that the cross product is distributive and anti-commutative. The key property to notice here is that if the sum of three vectors is zero, then there is a relationship between their cross products that we can exploit. Specifically, we are going to use this property to show that the cross products of each pair of vectors are equal.

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Solution

We start with the given equation:

$$\underline{a} + \underline{b} + \underline{c} = 0$$

This implies:

$$\underline{c} = - (\underline{a} + \underline{b})$$

We will now prove the required identities step by step.

Step 1: Take the cross product of both sides with $\underline{a}$

Take the cross product of $\underline{a}$ with both sides of the equation:

$$\underline{a} \times (\underline{a} + \underline{b} + \underline{c}) = \underline{a} \times 0$$

This simplifies to:

$$\underline{a} \times \underline{a} + \underline{a} \times \underline{b} + \underline{a} \times \underline{c} = 0$$

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

$$0 + \underline{a} \times \underline{b} + \underline{a} \times \underline{c} = 0$$

This simplifies to:

$$\underline{a} \times \underline{b} + \underline{a} \times \underline{c} = 0$$

Therefore:

$$\underline{a} \times \underline{b} = - \underline{a} \times \underline{c}$$

or equivalently:

$$\underline{a} \times \underline{b} = \underline{c} \times \underline{a}$$

which is the first part of the required result.

Step 2: Cross-multiply with $\underline{b}$

Now, take the cross product of $\underline{b}$ with both sides of the original equation:

$$\underline{b} \times (\underline{a} + \underline{b} + \underline{c}) = \underline{b} \times 0$$

This simplifies to:

$$\underline{b} \times \underline{a} + \underline{b} \times \underline{b} + \underline{b} \times \underline{c} = 0$$

Since $\underline{b} \times \underline{b} = 0$ (again, the cross product of any vector with itself is zero), we get:

$$\underline{b} \times \underline{a} + \underline{b} \times \underline{c} = 0$$

This simplifies to:

$$\underline{b} \times \underline{c} = - \underline{b} \times \underline{a}$$

or equivalently:

$$\underline{b} \times \underline{c} = \underline{a} \times \underline{b}$$

This is the second part of the required result.

Step 3: Combine the results

From Step 1 and Step 2, we have:

$$\underline{a} \times \underline{b} = \underline{c} \times \underline{a} \quad \text{and} \quad \underline{b} \times \underline{c} = \underline{a} \times \underline{b}$$

Thus, we can conclude:

$$\underline{a} \times \underline{b} = \underline{b} \times \underline{c} = \underline{c} \times \underline{a}$$

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Key Formulas or Methods Used

1. Distributive Property of the Cross Product:

$$\underline{a} \times (\underline{b} + \underline{c}) = \underline{a} \times \underline{b} + \underline{a} \times \underline{c}$$

2. Anti-commutative Property of the Cross Product:

$$\underline{a} \times \underline{b} = - \underline{b} \times \underline{a}$$

3. Cross Product of a Vector with Itself:

$$\underline{a} \times \underline{a} = 0$$

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Summary of Steps

1. Start with the given vector equation $\underline{a} + \underline{b} + \underline{c} = 0$.
2. Take the cross product of $\underline{a}$ with both sides of the equation and simplify.
3. Take the cross product of $\underline{b}$ with both sides of the equation and simplify.
4. Combine the results to show that $\underline{a} \times \underline{b} = \underline{b} \times \underline{c} = \underline{c} \times \underline{a}$.