Q.Find the value of $\lambda$ such that the vectors $\vec{a}=2\hat{i}+\lambda\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+3\hat{k}$ are orthogonal
(A) $0$
(B) $1$
(C) $\dfrac{3}{2}$
(D) $-\dfrac{5}{2}$

Concept: Perpendicular Vectors Condition — two vectors are orthogonal iff their dot product is zero.

Step 1: Write the dot product of $\vec{a}$ and $\vec{b}$:

Step 2: Simplify:

$\vec{a} \cdot \vec{b} = 2 + 2\lambda + 3 = 5 + 2\lambda$ …

The key idea here is the orthogonality condition for vectors. Two vectors are orthogonal (perpendicular) if and only if their dot product is zero. This is a fundamental geometric fact: the dot product measures how much two vectors point in the same direction. When it’s zero, they are at right angles.

So the problem reduces to a simple algebraic equation. Let’s go step by step.

1. Write the dot product explicitly. For $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$, the dot product is:

Multiply corresponding components and add.

2. Simplify the expression.

3. Set the dot product equal to zero (orthogonality condition).

4. Solve for $\lambda$. $2\lambda = -5 \quad \Rightarrow \quad \lambda = -\frac{5}{2}$ …

Method: Finding a Parameter That Makes Two Vectors Orthogonal

Use this when a vector contains an unknown and must be perpendicular to another.

Steps

Step 1: Apply the orthogonality condition

Perpendicular means $\vec{a}\cdot\vec{b}=0$ — not a zero cross product, and not proportional components (that would be parallel).

Step 2: Write the dot product with the unknown kept symbolic

Multiply corresponding components and sum, producing a linear expression in the unknown parameter. …

Common Mistakes

Mistake 1: Setting the cross product (or proportional components) to zero instead of the dot product

Why it's wrong: orthogonal means $\vec{a}\cdot\vec{b}=0$; a zero cross product or proportional components is the parallel condition. Correct approach: compute the dot product and set it to zero.

Mistake 2: Sign error solving $5+2\lambda=0$ …