Let a three- dimensional vector `vecV` satisfy the condition , `2vecV + vecV xx ( hati + 2hatj ) = 2hati + hatk` . If `3|vecV| = sqrtm` . Then find the value of m.

Explanation

To solve the given problem, we start from the equation provided:

**Given:**
\[ 2\vec{V} + \vec{V} \times (\hat{i} + 2\hat{j}) = 2\hat{i} + \hat{k} \]

**Step 1: Define the vector \(\vec{V}\)**

Let:
\[ \vec{V} = a\hat{i} + b\hat{j} + c\hat{k} \]

**Step 2: Calculate the cross product \(\vec{V} \times (\hat{i} + 2\hat{j})\)**

Using the determinant form for the cross product:
\[ \vec{V} \times (\hat{i} + 2\hat{j}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ 1 & 2 & 0 \end{vmatrix} \]

Calculating the determinant:
\[ = \hat{i}(b \cdot 0 - c \cdot 2) - \hat{j}(a \cdot 0 - c \cdot 1) + \hat{k}(a \cdot 2 - b \cdot 1) \]
\[ = -2c\hat{i} + c\hat{j} + (2a - b)\hat{k} \]

Thus,
\[ \vec{V} \times (\hat{i} + 2\hat{j}) = -2c\hat{i} + c\hat{j} + (2a - b)\hat{k} \]

**Step 3: Substitute back into the original equation**

Now substituting this back into the original equation:
\[ 2\vec{V} + \left(-2c\hat{i} + c\hat{j} + (2a - b)\hat{k}\right) = 2\hat{i} + \hat{k} \]
This gives:
\[ 2(a\hat{i} + b\hat{j} + c\hat{k}) + (-2c\hat{i} + c\hat{j} + (2a - b)\hat{k}) = 2\hat{i} + \hat{k} \]

Expanding this:
\[ (2a - 2c)\hat{i} + (2b + c)\hat{j} + (2c + 2a - b)\hat{k} = 2\hat{i} + \hat{k} \]

**Step 4: Equate coefficients**

From this, we can equate the coefficients of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\):
1. \(2a - 2c = 2\)
2. \(2b + c = 0\)
3. \(2c + 2a - b = 1\)

**Step 5: Solve the equations**

From the first equation:
\[ 2a - 2c = 2 \implies a - c = 1 \implies a = c + 1 \quad (1) \]

From the second equation:
\[ 2b + c = 0 \implies c = -2b \quad (2) \]

Substituting (2) into (1):
\[ a = -2b + 1 \quad (3) \]

Now substituting (2) and (3) into the third equation:
\[ 2(-2b) + 2(-2b + 1) - b = 1 \]
\[ -4b - 4b + 2 - b = 1 \]
\[ -9b + 2 = 1 \implies -9b = -1 \implies b = \frac{1}{9} \]

Now substituting \(b\) back into (2):
\[ c = -2 \times \frac{1}{9} = -\frac{2}{9} \]

Substituting \(b\) into (3):
\[ a = -2 \times \frac{1}{9} + 1 = 1 - \frac{2}{9} = \frac{7}{9} \]

Thus, we have:
\[ \vec{V} = \frac{7}{9}\hat{i} + \frac{1}{9}\hat{j} - \frac{2}{9}\hat{k} \]

**Step 6: Calculate the magnitude of \(\vec{V}\)**

The magnitude of \(\vec{V}\) is given by:
\[ |\vec{V}| = \sqrt{\left(\frac{7}{9}\right)^2 + \left(\frac{1}{9}\right)^2 + \left(-\frac{2}{9}\right)^2} \]
\[ = \sqrt{\frac{49}{81} + \frac{1}{81} + \frac{4}{81}} = \sqrt{\frac{54}{81}} = \sqrt{\frac{2}{3}} \]

**Step 7: Find \(m\)**

Given that \(3|\vec{V}| = \sqrt{m}\):
\[ 3 \cdot \sqrt{\frac{2}{3}} = \sqrt{m} \]
\[ \sqrt{m} = \sqrt{6} \implies m = 6 \]

**Final Answer:**
\[ m = 6 \]

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