If `|veca|=4, |vecb|=4` and `|vecc|=5` such that `veca bot vecb + vecc, vecb bot (vecc + veca)` and `vecc bot (veca + vecb)`, then `|veca + vecb + vecc|` is:

Options

A. 7
B. 5
C. 13
D. `sqrt(57)`

Answer

To solve the problem step by step, we will use the given conditions about the vectors and their magnitudes.

### Step 1: Understand the Given Information
We are given:
- Magnitude of vector \( \vec{a} \) is \( |\vec{a}| = 4 \)
- Magnitude of vector \( \vec{b} \) is \( |\vec{b}| = 4 \)
- Magnitude of vector \( \vec{c} \) is \( |\vec{c}| = 5 \)

Additionally, we know:
- \( \vec{a} \bot (\vec{b} + \vec{c}) \)
- \( \vec{b} \bot (\vec{c} + \vec{a}) \)
- \( \vec{c} \bot (\vec{a} + \vec{b}) \)

### Step 2: Set Up the Dot Product Equations
Since the vectors are perpendicular to the sums of the other two vectors, we can write the following equations using the dot product:
1. \( \vec{a} \cdot (\vec{b} + \vec{c}) = 0 \)
- This expands to \( \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \) (Equation 1)

2. \( \vec{b} \cdot (\vec{c} + \vec{a}) = 0 \)
- This expands to \( \vec{b} \cdot \vec{c} + \vec{b} \cdot \vec{a} = 0 \) (Equation 2)

3. \( \vec{c} \cdot (\vec{a} + \vec{b}) = 0 \)
- This expands to \( \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \) (Equation 3)

### Step 3: Analyze the Dot Product Equations
From these equations, we can deduce:
- From Equation 1: \( \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \) implies \( \vec{a} \cdot \vec{b} = -\vec{a} \cdot \vec{c} \)
- From Equation 2: \( \vec{b} \cdot \vec{c} + \vec{b} \cdot \vec{a} = 0 \) implies \( \vec{b} \cdot \vec{c} = -\vec{b} \cdot \vec{a} \)
- From Equation 3: \( \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \) implies \( \vec{c} \cdot \vec{a} = -\vec{c} \cdot \vec{b} \)

### Step 4: Calculate the Magnitude of \( \vec{a} + \vec{b} + \vec{c} \)
To find \( |\vec{a} + \vec{b} + \vec{c}| \), we calculate:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} + \vec{b} + \vec{c}) \]
Expanding this gives:
\[ |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \]
Substituting the magnitudes:
\[ = 4^2 + 4^2 + 5^2 + 2(0) = 16 + 16 + 25 + 0 = 57 \]

### Step 5: Final Calculation
Thus, we have:
\[ |\vec{a} + \vec{b} + \vec{c}|^2 = 57 \]
Taking the square root:
\[ |\vec{a} + \vec{b} + \vec{c}| = \sqrt{57} \]

### Final Answer
The magnitude \( |\vec{a} + \vec{b} + \vec{c}| \) is \( \sqrt{57} \).

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