Q.The value of $\lambda$ for which the vectors $3\hat{i}-6\hat{j}+\hat{k}$ and $2\hat{i}-4\hat{j}+\lambda\hat{k}$ are parallel is
(A) $\dfrac{2}{3}$
(B) $\dfrac{3}{2}$
(C) $\dfrac{5}{2}$
(D) $\dfrac{2}{5}$

Concept: Parallel Vectors Condition — Two vectors are parallel if one is a scalar multiple of the other; their corresponding components must be proportional.

Step 1: Let $\vec{a} = 3\hat{i} - 6\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - 4\hat{j} + \lambda\hat{k}$. For parallelism, there exists a scalar $k$ such that $\vec{a} = k\vec{b}$.

Step 2: Equate components:

• $\hat{i}$: $3 = k(2) \Rightarrow k = \frac{3}{2}$ …

Why the “parallel vectors” condition works

When two vectors are parallel, they point in exactly the same (or exactly opposite) direction. That means one vector is just a stretched or shrunk version of the other — a scalar multiple. So if $\vec{a}$ and $\vec{b}$ are parallel, there exists some real number $k$ such that:

This is the cleanest way to handle the problem. No dot products, no cross products — just component-wise equality.

Step-by-step solution

1. Write the vectors clearly

Let

$\vec{a} = 3\hat{i} - 6\hat{j} + \hat{k}$

$\vec{b} = 2\hat{i} - 4\hat{j} + \lambda\hat{k}$

2. Set up the scalar multiple condition

If $\vec{a}$ and $\vec{b}$ are parallel, then:

for some scalar $k$. Writing this component-wise:

3. Equate the $\hat{i}$ components

From the $\hat{i}$ coefficients:

4. Check consistency with the $\hat{j}$ components

From the $\hat{j}$ coefficients:

Substitute $k = \frac{2}{3}$:

This holds true — so the $\hat{i}$ and $\hat{j}$ components are consistent. That confirms our $k$ is correct. …

Method: Testing two vectors for parallelism by proportional components

Use this whenever a question asks for an unknown so that two given vectors are parallel (or collinear).

Steps

Step 1: State the parallel condition as a scalar multiple

Two vectors are parallel exactly when one is a scalar multiple of the other. So write $\vec{a}=k\,\vec{b}$ (or $\vec{b}=k\,\vec{a}$) for some unknown scalar $k$.

Step 2: Turn it into proportional components

Equating $\hat{i},\hat{j},\hat{k}$ coefficients gives the component-ratio form:

Step 3: Find the scalar from a fully-known ratio …

Common Mistakes

Mistake 1: Reading the scalar $k$ off as the answer for $\lambda$

Why it's wrong: the ratio here is $\tfrac{3}{2}=\tfrac{-6}{-4}=\tfrac{1}{\lambda}$. Students find $k=\tfrac{3}{2}$ from the first components and wrongly report $\lambda=\tfrac{3}{2}$. Correct approach: $\lambda$ sits in the ratio $\tfrac{1}{\lambda}$, so $\tfrac{1}{\lambda}=\tfrac{3}{2}\Rightarrow\lambda=\tfrac{2}{3}$ — invert correctly.

Mistake 2: Setting up the ratio upside-down …