If `veca and vecb` are vectors such that `|veca+ecb|=sqrt(29) and veca xx (2hati+3hatj+4hatk)=(2hati+3hatj+4hatk) xxvecb,` then possible value of `(veca+vecb).(-7hati+2hatj+3hatk)` is (A) 0 (B) 3 (C) 4 (D) 8

Options

A. 0
B. 3
C. 4
D. 8

Answer

To solve the problem step by step, we will follow the information given in the question and the conditions provided.

### Step 1: Understand the Given Conditions
We know that:
1. \(|\vec{a} + \vec{b}| = \sqrt{29}\)
2. \(\vec{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{b}\)

### Step 2: Analyze the Cross Product Condition
From the second condition, we can use the property of cross products. If \( \vec{m} \times \vec{n} = \vec{n} \times \vec{p} \), then it implies that:
\[ \vec{m} = k \vec{n} \quad \text{and} \quad \vec{p} = k \vec{n} \]
for some scalar \( k \).

Here, let:
\[ \vec{n} = (2\hat{i} + 3\hat{j} + 4\hat{k}) \]
Thus, we can write:
\[ \vec{a} = k(2\hat{i} + 3\hat{j} + 4\hat{k}) \quad \text{and} \quad \vec{b} = k(2\hat{i} + 3\hat{j} + 4\hat{k}) \]

### Step 3: Calculate \(\vec{a} + \vec{b}\)
Now, we can find:
\[ \vec{a} + \vec{b} = k(2\hat{i} + 3\hat{j} + 4\hat{k}) + k(2\hat{i} + 3\hat{j} + 4\hat{k}) = 2k(2\hat{i} + 3\hat{j} + 4\hat{k}) \]

### Step 4: Find the Magnitude
The magnitude of \(\vec{a} + \vec{b}\) is given as:
\[ |\vec{a} + \vec{b}| = |2k(2\hat{i} + 3\hat{j} + 4\hat{k})| = 2k \sqrt{(2^2 + 3^2 + 4^2)} = 2k \sqrt{29} \]
Setting this equal to \(\sqrt{29}\):
\[ 2k \sqrt{29} = \sqrt{29} \]
Dividing both sides by \(\sqrt{29}\) (assuming \(\sqrt{29} \neq 0\)):
\[ 2k = 1 \implies k = \frac{1}{2} \]

### Step 5: Substitute Back to Find \(\vec{a}\) and \(\vec{b}\)
Now substituting \(k\) back:
\[ \vec{a} = \frac{1}{2}(2\hat{i} + 3\hat{j} + 4\hat{k}) = \hat{i} + \frac{3}{2}\hat{j} + 2\hat{k} \]
\[ \vec{b} = \frac{1}{2}(2\hat{i} + 3\hat{j} + 4\hat{k}) = \hat{i} + \frac{3}{2}\hat{j} + 2\hat{k} \]

### Step 6: Calculate \(\vec{a} + \vec{b}\)
Thus,
\[ \vec{a} + \vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k} \]

### Step 7: Find the Dot Product
Now we need to calculate:
\[ (\vec{a} + \vec{b}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) \]
Calculating the dot product:
\[ = (2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}) \]
\[ = 2 \cdot (-7) + 3 \cdot 2 + 4 \cdot 3 \]
\[ = -14 + 6 + 12 \]
\[ = -14 + 18 = 4 \]

### Final Answer
The possible value of \((\vec{a} + \vec{b}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k})\) is \(4\).