Question

The value of $\lambda$ such that the vectors $2\hat{i}-\hat{j}+2\hat{k}$ and $3\hat{i}+2\lambda\hat{j}$ are perpendicular is

• A. 0
• B. 1
• C. 2
• D. 3
• E. 4

Hint:

Vector Tip: Whenever a question says "perpendicular" or "orthogonal," immediately set the dot product to zero. It is the most reliable and common property tested in vector algebra!

Answer 1

The Correct Option is D

Concept:
Two vectors $\vec{u}$ and $\vec{v}$ are orthogonal (perpendicular) if and only if their dot product is equal to zero ($\vec{u} \cdot \vec{v} = 0$). The dot product is calculated by multiplying corresponding components and summing them.

Step 1: Identify the components of the vectors.
Let $\vec{u} = 2\hat{i} - \hat{j} + 2\hat{k}$. Its components are $(2, -1, 2)$. Let $\vec{v} = 3\hat{i} + 2\lambda\hat{j} + 0\hat{k}$. Its components are $(3, 2\lambda, 0)$.

Step 2: Set up the dot product equation.
Since the vectors are perpendicular, we set $\vec{u} \cdot \vec{v} = 0$: $$(2)(3) + (-1)(2\lambda) + (2)(0) = 0$$

Step 3: Simplify the equation.
Perform the multiplications: $$6 - 2\lambda + 0 = 0$$ $$6 - 2\lambda = 0$$

Step 4: Isolate the variable term.
Add $2\lambda$ to both sides of the equation: $$6 = 2\lambda$$

Step 5: Solve for the unknown scalar $\lambda$.
Divide both sides by 2: $$\lambda = \frac{6}{2} = 3$$ Hence the correct answer is (D) 3.