Let `veca=2hati+3hatj+4hatk, vecb=hati-2hatj+hatk` and `vecc=hati+hatj-hatk.` If `vecr xx veca =vecb` and `vecr.vec c=3,` then the value of `|vecr|` is equal to

Options

A. `sqrt(155)`
B. `sqrt(17)`
C. `2sqrt(17)`
D. 3

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and derive the solution systematically.

### Given:
- \(\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}\)
- \(\vec{b} = \hat{i} - 2\hat{j} + \hat{k}\)
- \(\vec{c} = \hat{i} + \hat{j} - \hat{k}\)
- \(\vec{r} \times \vec{a} = \vec{b}\)
- \(\vec{r} \cdot \vec{c} = 3\)

### Step 1: Use the vector triple product identity
We know that:
\[ \vec{r} \times \vec{a} = \vec{b} \]
Taking the dot product of both sides with \(\vec{c}\):
\[ \vec{c} \cdot (\vec{r} \times \vec{a}) = \vec{c} \cdot \vec{b} \]

Using the vector triple product identity:
\[ \vec{c} \cdot (\vec{r} \times \vec{a}) = (\vec{r} \cdot \vec{c}) \vec{a} - (\vec{a} \cdot \vec{c}) \vec{r} \]

### Step 2: Substitute known values
From the problem, we know:
\[ \vec{r} \cdot \vec{c} = 3 \]
Thus:
\[ 3\vec{a} - (\vec{a} \cdot \vec{c}) \vec{r} = \vec{c} \cdot \vec{b} \]

### Step 3: Calculate \(\vec{a} \cdot \vec{c}\)
\[ \vec{a} \cdot \vec{c} = (2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (\hat{i} + \hat{j} - \hat{k}) = 2 \cdot 1 + 3 \cdot 1 + 4 \cdot (-1) = 2 + 3 - 4 = 1 \]

### Step 4: Substitute \(\vec{a} \cdot \vec{c}\) into the equation
Now substituting \(\vec{a} \cdot \vec{c} = 1\):
\[ 3\vec{a} - 1\vec{r} = \vec{c} \cdot \vec{b} \]

### Step 5: Calculate \(\vec{c} \cdot \vec{b}\)
\[ \vec{c} \cdot \vec{b} = (\hat{i} + \hat{j} - \hat{k}) \cdot (\hat{i} - 2\hat{j} + \hat{k}) = 1 \cdot 1 + 1 \cdot (-2) + (-1) \cdot 1 = 1 - 2 - 1 = -2 \]

### Step 6: Substitute \(\vec{c} \cdot \vec{b}\) into the equation
Now we have:
\[ 3\vec{a} - \vec{r} = -2 \]
Thus:
\[ \vec{r} = 3\vec{a} + 2 \]

### Step 7: Calculate \(3\vec{a}\)
\[ 3\vec{a} = 3(2\hat{i} + 3\hat{j} + 4\hat{k}) = 6\hat{i} + 9\hat{j} + 12\hat{k} \]

### Step 8: Solve for \(\vec{r}\)
\[ \vec{r} = (6 + 2)\hat{i} + 9\hat{j} + 12\hat{k} = 8\hat{i} + 9\hat{j} + 12\hat{k} \]

### Step 9: Calculate the magnitude of \(\vec{r}\)
\[ |\vec{r}| = \sqrt{(8)^2 + (9)^2 + (12)^2} = \sqrt{64 + 81 + 144} = \sqrt{289} = 17 \]

### Final Answer
The value of \(|\vec{r}|\) is \(17\).