Q.(a) If vectors 𝑎⃗ = 2ı̂ + 2ȷ̂ + 3k̂ , 𝑏⃗⃗ = − ı̂ + 2ȷ̂ + k̂ and 𝑐⃗ = 3ı̂ + ȷ̂ are such that 𝑏⃗⃗ + λ𝑐⃗ is perpendicular to 𝑎⃗ , then find the value of λ. OR

Two vectors are perpendicular exactly when their dot product is zero, so set $(\vec{b}+\lambda\vec{c})\cdot\vec{a}=0$.

With $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$, $\vec{c}=3\hat{i}+\hat{j}$:

$\vec{b}+\lambda\vec{c}=(-1+3\lambda)\hat{i}+(2+\lambda)\hat{j}+\hat{k}$

Dot with $\vec{a}$: …

The idea

Two vectors are perpendicular precisely when their dot product is zero. We are told $\vec{b}+\lambda\vec{c}$ is perpendicular to $\vec{a}$, so we build that combined vector, dot it with $\vec{a}$, set the result to $0$, and solve the resulting linear equation for $\lambda$.

Set up the vectors

Form $\vec{b}+\lambda\vec{c}$

Add component by component (note $\vec{c}$ has no $\hat{k}$ part):

Apply the perpendicularity condition …

Method: Solving for an unknown scalar using the perpendicularity condition

Use this whenever a problem states one vector (often containing an unknown like $\lambda$) is perpendicular to another and asks you to find that unknown.

Steps

Step 1: Translate "perpendicular" into a dot product of zero.

The defining test is

Seeing the word "perpendicular" should immediately trigger "set the dot product to $0$" — not equal magnitudes, not a cross product.

Step 2: Build the compound vector, keeping the unknown symbolic. …

Common Mistakes

Mistake 1: Forgetting that $\vec{c}=3\hat{i}+\hat{j}$ has zero $\hat{k}$-component.

Why it's wrong: treating a missing component as anything but $0$ corrupts $\vec{b}+\lambda\vec{c}$ and the dot product. Correct approach: write $\vec{c}=3\hat{i}+\hat{j}+0\hat{k}$ explicitly.

Mistake 2: Setting magnitudes equal instead of the dot product to zero.

Why it's wrong: perpendicularity is $(\vec{b}+\lambda\vec{c})\cdot\vec{a}=0$, not $|\vec{b}+\lambda\vec{c}|=|\vec{a}|$. Correct approach: reach for the dot-product-zero condition whenever "perpendicular" appears. …