The value of `lambda` for which the vector `3 hat (i) -6 hat(j) + hat(k)` and `2 hat(i) - 4 hat(j) + lambda vec(k)` are parallel is

Options

A. `(3)/(2)`
B. `(2)/(3)`
C. `(5)/(2)`
D. `(2)/(5)`

Answer

To determine the value of \( \lambda \) for which the vectors \( \vec{A} = 3\hat{i} - 6\hat{j} + \hat{k} \) and \( \vec{B} = 2\hat{i} - 4\hat{j} + \lambda\hat{k} \) are parallel, we can follow these steps:

### Step 1: Identify the components of the vectors
The components of the vectors are:
- For \( \vec{A} \): \( a_1 = 3 \), \( b_1 = -6 \), \( c_1 = 1 \)
- For \( \vec{B} \): \( a_2 = 2 \), \( b_2 = -4 \), \( c_2 = \lambda \)

### Step 2: Set up the condition for parallel vectors
Two vectors are parallel if the ratios of their corresponding components are equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
This gives us the following equations:
\[ \frac{3}{2} = \frac{-6}{-4} = \frac{1}{\lambda} \]

### Step 3: Solve the first ratio
We can simplify the second ratio:
\[ \frac{-6}{-4} = \frac{6}{4} = \frac{3}{2} \]
Thus, we have:
\[ \frac{3}{2} = \frac{1}{\lambda} \]

### Step 4: Cross-multiply to find \( \lambda \)
Cross-multiplying gives:
\[ 3\lambda = 2 \]
Now, solve for \( \lambda \):
\[ \lambda = \frac{2}{3} \]

### Step 5: Verify using the third ratio
We can also check using the third ratio:
\[ \frac{3}{2} = \frac{1}{\lambda} \]
Cross-multiplying gives the same result:
\[ 3\lambda = 2 \implies \lambda = \frac{2}{3} \]

### Conclusion
The value of \( \lambda \) for which the vectors are parallel is:
\[ \lambda = \frac{2}{3} \]

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