If `vecr*hati=2vecr*hatj=4hatr*hatk and |vecr|=sqrt(84),` then the value of `vecr*(2hati-3hatj+hatk)` may be

Options

A. 0
B. 2
C. 4
D. 6

Answer

To solve the problem, we need to find the value of the dot product of the vector \(\vec{r}\) with the vector \(2\hat{i} - 3\hat{j} + \hat{k}\).

### Step-by-Step Solution:

1. **Define the Vector \(\vec{r}\)**:
Let \(\vec{r} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\).

2. **Use the Given Conditions**:
From the problem, we have:
\[ \vec{r} \cdot \hat{i} = a_1, \]
\[ 2 \vec{r} \cdot \hat{j} = 2a_2, \]
\[ 4 \vec{r} \cdot \hat{k} = 4a_3. \]
This implies:
\[ a_1 = 2a_2 = 4a_3. \]

3. **Express \(a_2\) and \(a_3\) in terms of \(a_1\)**:
From \(a_1 = 2a_2\), we can express \(a_2\) as:
\[ a_2 = \frac{a_1}{2}. \]
From \(a_1 = 4a_3\), we can express \(a_3\) as:
\[ a_3 = \frac{a_1}{4}. \]

4. **Use the Magnitude Condition**:
We know that the magnitude of \(\vec{r}\) is given as \(|\vec{r}| = \sqrt{84}\):
\[ |\vec{r}| = \sqrt{a_1^2 + a_2^2 + a_3^2} = \sqrt{84}. \]
Substituting \(a_2\) and \(a_3\):
\[ \sqrt{a_1^2 + \left(\frac{a_1}{2}\right)^2 + \left(\frac{a_1}{4}\right)^2} = \sqrt{84}. \]
Squaring both sides:
\[ a_1^2 + \frac{a_1^2}{4} + \frac{a_1^2}{16} = 84. \]

5. **Combine the Terms**:
To combine the terms, find a common denominator (16):
\[ \frac{16a_1^2}{16} + \frac{4a_1^2}{16} + \frac{a_1^2}{16} = 84, \]
\[ \frac{21a_1^2}{16} = 84. \]
Multiplying both sides by 16:
\[ 21a_1^2 = 1344. \]
Dividing by 21:
\[ a_1^2 = \frac{1344}{21} = 64. \]
Hence,
\[ a_1 = \pm 8. \]

6. **Find \(a_2\) and \(a_3\)**:
If \(a_1 = 8\):
\[ a_2 = \frac{8}{2} = 4, \quad a_3 = \frac{8}{4} = 2. \]
If \(a_1 = -8\):
\[ a_2 = \frac{-8}{2} = -4, \quad a_3 = \frac{-8}{4} = -2. \]

7. **Express \(\vec{r}\)**:
Therefore, we have two possible vectors:
\[ \vec{r} = 8\hat{i} + 4\hat{j} + 2\hat{k} \quad \text{or} \quad \vec{r} = -8\hat{i} - 4\hat{j} - 2\hat{k}. \]

8. **Calculate the Dot Product**:
We need to find \(\vec{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k})\):
- For \(\vec{r} = 8\hat{i} + 4\hat{j} + 2\hat{k}\):
\[ \vec{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k}) = 8 \cdot 2 + 4 \cdot (-3) + 2 \cdot 1 = 16 - 12 + 2 = 6. \]
- For \(\vec{r} = -8\hat{i} - 4\hat{j} - 2\hat{k}\):
\[ \vec{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k}) = -8 \cdot 2 + (-4) \cdot (-3) + (-2) \cdot 1 = -16 + 12 - 2 = -6. \]

### Final Result:
The value of \(\vec{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k})\) can be either \(6\) or \(-6\).