Let `veca= 2hati -7hatj+5hat k, vecb=hati+hat k and vec c =hat i +2hat j -3hat k` be three given vectors. If `vec r` is a vector such that `vec r times vec a= vec c times vec a and vec r .vec b=0,` then `abs(vec r)` is

Options

A. `(11)/7 sqrt2`
B. `11/7`
C. `(sqrt914)/7`
D. `(11)/5sqrt2`

Answer

To solve the problem, we need to find the magnitude of the vector \(\vec{r}\) given the conditions involving vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

### Step 1: Write down the given vectors
We have:
\[ \vec{a} = 2\hat{i} - 7\hat{j} + 5\hat{k} \]
\[ \vec{b} = \hat{i} + \hat{k} \]
\[ \vec{c} = \hat{i} + 2\hat{j} - 3\hat{k} \]

### Step 2: Set up the equations
We know that:
\[ \vec{r} \times \vec{a} = \vec{c} \times \vec{a} \]
This implies:
\[ \vec{r} - \vec{c} = \lambda \vec{a} \]
for some scalar \(\lambda\). Thus, we can express \(\vec{r}\) as:
\[ \vec{r} = \vec{c} + \lambda \vec{a} \]

### Step 3: Use the second condition
The second condition given is:
\[ \vec{r} \cdot \vec{b} = 0 \]
Substituting \(\vec{r}\) into this equation gives:
\[ (\vec{c} + \lambda \vec{a}) \cdot \vec{b} = 0 \]

### Step 4: Calculate \(\vec{c} \cdot \vec{b}\) and \(\vec{a} \cdot \vec{b}\)
Calculating \(\vec{c} \cdot \vec{b}\):
\[ \vec{c} \cdot \vec{b} = (\hat{i} + 2\hat{j} - 3\hat{k}) \cdot (\hat{i} + \hat{k}) = 1 \cdot 1 + 2 \cdot 0 - 3 \cdot 1 = 1 - 3 = -2 \]

Calculating \(\vec{a} \cdot \vec{b}\):
\[ \vec{a} \cdot \vec{b} = (2\hat{i} - 7\hat{j} + 5\hat{k}) \cdot (\hat{i} + \hat{k}) = 2 \cdot 1 + (-7) \cdot 0 + 5 \cdot 1 = 2 + 5 = 7 \]

### Step 5: Substitute back into the equation
Now substituting back into the equation:
\[ -2 + \lambda \cdot 7 = 0 \]
Solving for \(\lambda\):
\[ \lambda \cdot 7 = 2 \implies \lambda = \frac{2}{7} \]

### Step 6: Substitute \(\lambda\) back into \(\vec{r}\)
Now substituting \(\lambda\) back into the expression for \(\vec{r}\):
\[ \vec{r} = \vec{c} + \frac{2}{7} \vec{a} = \left(\hat{i} + 2\hat{j} - 3\hat{k}\right) + \frac{2}{7}(2\hat{i} - 7\hat{j} + 5\hat{k}) \]
Calculating this gives:
\[ \vec{r} = \hat{i} + 2\hat{j} - 3\hat{k} + \left(\frac{4}{7}\hat{i} - 2\hat{j} + \frac{10}{7}\hat{k}\right) \]
Combining the components:
\[ \vec{r} = \left(1 + \frac{4}{7}\right)\hat{i} + \left(2 - 2\right)\hat{j} + \left(-3 + \frac{10}{7}\right)\hat{k} \]
\[ = \frac{11}{7}\hat{i} + 0\hat{j} + \left(-\frac{21}{7} + \frac{10}{7}\right)\hat{k} = \frac{11}{7}\hat{i} - \frac{11}{7}\hat{k} \]

### Step 7: Calculate the magnitude of \(\vec{r}\)
The magnitude of \(\vec{r}\) is given by:
\[ |\vec{r}| = \sqrt{\left(\frac{11}{7}\right)^2 + 0^2 + \left(-\frac{11}{7}\right)^2} = \sqrt{\frac{121}{49} + \frac{121}{49}} = \sqrt{\frac{242}{49}} = \frac{\sqrt{242}}{7} = \frac{11\sqrt{2}}{7} \]

### Final Answer
Thus, the magnitude of \(\vec{r}\) is:
\[ |\vec{r}| = \frac{11\sqrt{2}}{7} \]