Q.Show that the vectors $2\hat{i} - 3\hat{j} + 4\hat{k}$ and $-4\hat{i} + 6\hat{j} - 8\hat{k}$ are collinear.

Concept: Collinear Vectors Property — Two vectors are collinear if one is a scalar multiple of the other.

Step 1: Let $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\vec{b} = -4\hat{i} + 6\hat{j} - 8\hat{k}$.

Step 2: Check if $\vec{b} = \lambda \vec{a}$ for some scalar $\lambda$. Comparing components:

• $\hat{i}$: $-4 = \lambda (2) \implies \lambda = -2$
• $\hat{j}$: $6 = \lambda (-3) \implies \lambda = -2$ …

Why this works

Collinear vectors lie along the same line — they have the same direction (or exactly opposite direction). In component form, this means each corresponding component of one vector is proportional to the component of the other, with the same constant of proportionality for all three components. If that constant exists, the vectors are scalar multiples of each other.

Let’s check.

1. Write the vectors clearly

2. Look for a scalar $k$ such that $\vec{b} = k\vec{a}$ Compare the $\hat{i}$ components:

3. Check the $\hat{j}$ component

Matches.

4. Check the $\hat{k}$ component

Matches.

Since the same $k = -2$ works for all three components, $\vec{b} = -2\vec{a}$. …

Method: Testing Two Vectors for Collinearity

Use this whenever you must show two vectors are collinear (parallel/anti-parallel) — or decide that they are not.

Steps

Step 1: State the test.

Two vectors $\vec{a}$ and $\vec{b}$ are collinear if and only if one is a scalar multiple of the other:

Step 2: Solve for $\lambda$ from one component.

Pick a convenient component and read off $\lambda$, e.g. from the $\hat{i}$ terms $b_1=\lambda a_1\Rightarrow\lambda=\dfrac{b_1}{a_1}$.

Step 3: Verify the SAME $\lambda$ works for every component. …

Common Mistakes

Mistake 1: Checking only one component.

Why it's wrong: matching $\lambda$ on the $\hat{i}$ terms alone does not prove collinearity — all three ratios must be equal. Correct approach: confirm the same $\lambda$ satisfies the $\hat{i}$, $\hat{j}$ and $\hat{k}$ components.

Mistake 2: Concluding "not collinear" because the vectors point opposite ways. …