7.4 Q-9

Question Statement

Given that:

$$\underline{a} \times \underline{b} = 0 \quad \text{and} \quad \underline{a} \cdot \underline{b} = 0$$

What conclusion can be drawn about the vectors $\underline{a}$ or $\underline{b}$?

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Background and Explanation

In vector analysis:

• Cross product ($\times$) results in a vector perpendicular to the two vectors involved. If the cross product is zero, it implies that the two vectors are either parallel or one of them is a zero vector.
• Dot product ($\cdot$) results in a scalar value that is zero when two vectors are perpendicular.

To solve this problem, we will use these properties of the cross product and dot product to draw a conclusion about the relationship between the two vectors.

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Solution

Step 1: Analyze the cross product condition

The given condition is:

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

This means that the vectors $\underline{a}$ and $\underline{b}$ are either parallel or one of them is the zero vector. This is because the cross product of two vectors is zero if and only if they are parallel (or one of the vectors is the zero vector).

Step 2: Analyze the dot product condition

The next condition is:

$$\underline{a} \cdot \underline{b} = 0$$

This implies that $\underline{a}$ and $\underline{b}$ are perpendicular to each other because the dot product is zero when the angle between the two vectors is $90^\circ$.

Step 3: Resolve the contradiction

Now, we have a contradiction. From Step 1, we know that $\underline{a}$ and $\underline{b}$ must be parallel, and from Step 2, we know they must be perpendicular. This is impossible because two vectors cannot be both parallel and perpendicular to each other simultaneously.

Step 4: Conclusion

The only way to resolve this contradiction is if at least one of the vectors is the zero vector. This would make both the cross product and dot product conditions true simultaneously.

Therefore, we can conclude that:

• At least one of the vectors $\underline{a}$ or $\underline{b}$ must be the zero vector.

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Key Formulas or Methods Used

1. Cross Product:

   • If $\underline{a} \times \underline{b} = 0$, then $\underline{a}$ and $\underline{b}$ are parallel or one of them is the zero vector.

2. Dot Product:

   • If $\underline{a} \cdot \underline{b} = 0$, then $\underline{a}$ and $\underline{b}$ are perpendicular.

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Summary of Steps

1. The cross product condition $\underline{a} \times \underline{b} = 0$ implies that $\underline{a}$ and $\underline{b}$ are parallel (or one of the vectors is the zero vector).
2. The dot product condition $\underline{a} \cdot \underline{b} = 0$ implies that $\underline{a}$ and $\underline{b}$ are perpendicular.
3. Since both conditions cannot be true for non-zero vectors, we conclude that at least one of the vectors must be the zero vector.