If `vecrxxveca=vecrxxvecb , veca=2hati-3hatj+4hatk, vecb=7hati+hatj-6hatk,vecc=hati+2hatj+hatk , and vecr*vecc=-3` then find `vecr*veca`

Answer

To solve the problem, we need to find the dot product \( \vec{r} \cdot \vec{a} \) given the conditions provided. Let's break it down step by step.

### Step 1: Understand the Given Information
We have:
- \( \vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k} \)
- \( \vec{b} = 7\hat{i} + \hat{j} - 6\hat{k} \)
- \( \vec{c} = \hat{i} + 2\hat{j} + \hat{k} \)
- \( \vec{r} \cdot \vec{c} = -3 \)
- \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b} \)

### Step 2: Use the Cross Product Condition
From the condition \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b} \), we can rearrange it as:
\[ \vec{r} \times \vec{a} - \vec{r} \times \vec{b} = \vec{0} \]
This implies:
\[ \vec{r} \times (\vec{a} - \vec{b}) = \vec{0} \]
This means that \( \vec{r} \) is parallel to \( \vec{a} - \vec{b} \).

### Step 3: Calculate \( \vec{a} - \vec{b} \)
Now, we compute \( \vec{a} - \vec{b} \):
\[ \vec{a} - \vec{b} = (2\hat{i} - 3\hat{j} + 4\hat{k}) - (7\hat{i} + \hat{j} - 6\hat{k}) \]
\[ = (2 - 7)\hat{i} + (-3 - 1)\hat{j} + (4 + 6)\hat{k} \]
\[ = -5\hat{i} - 4\hat{j} + 10\hat{k} \]

### Step 4: Express \( \vec{r} \)
Since \( \vec{r} \) is parallel to \( \vec{a} - \vec{b} \), we can write:
\[ \vec{r} = \lambda (-5\hat{i} - 4\hat{j} + 10\hat{k}) \]
for some scalar \( \lambda \).

### Step 5: Use the Dot Product with \( \vec{c} \)
Now, we use the condition \( \vec{r} \cdot \vec{c} = -3 \):
\[ \vec{r} \cdot \vec{c} = \lambda (-5\hat{i} - 4\hat{j} + 10\hat{k}) \cdot (\hat{i} + 2\hat{j} + \hat{k}) \]
Calculating the dot product:
\[ = \lambda \left[ -5(1) + (-4)(2) + 10(1) \right] \]
\[ = \lambda \left[ -5 - 8 + 10 \right] \]
\[ = \lambda (-3) \]
Setting this equal to -3:
\[ \lambda (-3) = -3 \implies \lambda = 1 \]

### Step 6: Find \( \vec{r} \)
Thus, we have:
\[ \vec{r} = -5\hat{i} - 4\hat{j} + 10\hat{k} \]

### Step 7: Calculate \( \vec{r} \cdot \vec{a} \)
Now we can find \( \vec{r} \cdot \vec{a} \):
\[ \vec{r} \cdot \vec{a} = (-5\hat{i} - 4\hat{j} + 10\hat{k}) \cdot (2\hat{i} - 3\hat{j} + 4\hat{k}) \]
Calculating the dot product:
\[ = (-5)(2) + (-4)(-3) + (10)(4) \]
\[ = -10 + 12 + 40 \]
\[ = 42 \]

### Final Answer
Thus, the value of \( \vec{r} \cdot \vec{a} \) is \( \boxed{42} \).