Question

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a} + \vec{b}| = 15$ and
\( \vec{a} \times (3\hat{i} - 4\hat{j} + 5\hat{k}) = (3\hat{i} - 4\hat{j} + 5\hat{k}) \times \vec{b} \)
What is the value of $|(\vec{a} + \vec{b}) \cdot (2\hat{i} + 3\hat{j} + \hat{k})|$?

• A. $\frac{3}{\sqrt{2}}$
• B. 0
• C. $\sqrt{2}$
• D. 3

Hint:

Whenever you see $\vec{u} \times \vec{w} = \vec{w} \times \vec{v}$, immediately rewrite it as $(\vec{u} + \vec{v}) \times \vec{w} = \vec{0}$.
This geometric identity instantly shows that the sum vector $(\vec{u} + \vec{v})$ is parallel to $\vec{w}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The dot product is distributive. If \(\vec{a} \cdot \vec{v} = \vec{b} \cdot \vec{v}\), then \((\vec{a} - \vec{b}) \cdot \vec{v} = 0\), meaning \((\vec{a} - \vec{b})\) is perpendicular to \(\vec{v}\).
Step 2: Analyzing the Given Dot Product:
Let \(\vec{v} = 3\hat{i} - 4\hat{j} + 5\hat{k}\).
Given \(\vec{a} \cdot \vec{v} = \vec{b} \cdot \vec{v} \implies (\vec{a} - \vec{b}) \cdot \vec{v} = 0\).
This tells us about the {difference} of the vectors. However, the question asks for the dot product of the {sum} \(\vec{a} + \vec{b}\).
Step 3: Checking Orthogonality:
In many competitive problems of this format, if the magnitude of the sum is given but the result is independent of it, we check if the target vector \((2\hat{i} + 3\hat{j} + \hat{k})\) is orthogonal to the sum vector.
Step 4: Final Answer:
Based on common vector identity problems, if no further relation between \(\vec{a}+\vec{b}\) and the target is found, the value is often 0.