Question

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be vectors of magnitude 2, 3 and 4 respectively. If $\vec{a} \cdot (\vec{b} + \vec{c}) = 0$, $\vec{b} \cdot (\vec{c} + \vec{a}) = 0$ and $\vec{c} \cdot (\vec{a} + \vec{b}) = 0$, then the magnitude of $\vec{a} + \vec{b} + \vec{c}$ is ______.

• A. 29
• B. $\sqrt{28}$
• C. $\sqrt{29}$
• D. 28

Hint:

If every vector is perfectly perpendicular to the sum of the other two vectors, it means all the mixed dot products sum to absolutely zero! The magnitude of their sum is just the square root of the sum of their individual squares ($\sqrt{A^2+B^2+C^2}$).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The square of the magnitude of the sum of vectors is $|\vec{a}+\vec{b}+\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a})$.

Step 2: Formula Application:
The given conditions: 1. $\vec{a}\cdot\vec{b} + \vec{a}\cdot\vec{c} = 0$ 2. $\vec{b}\cdot\vec{c} + \vec{b}\cdot\vec{a} = 0$ 3. $\vec{c}\cdot\vec{a} + \vec{c}\cdot\vec{b} = 0$

Step 3: Explanation:
Adding all three equations: $2(\vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a}) = 0$. So, $|\vec{a}+\vec{b}+\vec{c}|^2 = 2^2 + 3^2 + 4^2 + 0$ $|\vec{a}+\vec{b}+\vec{c}|^2 = 4 + 9 + 16 = 29$. $|\vec{a}+\vec{b}+\vec{c}| = \sqrt{29}$.

Step 4: Final Answer:
The magnitude is $\sqrt{29}$.