Question

The value of $\lambda$ for which 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 is

• A. $\frac{5}{2}$
• B. $\frac{-5}{2}$
• C. $\frac{2}{5}$
• D. $\frac{-2}{5}$

Hint:

"Orthogonal" is just a formal term for "perpendicular." Whenever you see it in a vector problem, immediately write down $\vec{u} \cdot \vec{v} = 0$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Two non-zero vectors are defined as orthogonal (perpendicular to each other) if and only if their scalar (dot) product is exactly zero.

Step 2: Key Formula or Approach:
For vectors $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, the orthogonality condition is: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 = 0$. Calculate the dot product of the given vectors and solve for the unknown parameter $\lambda$.

Step 3: Detailed Explanation:
Given vectors: $\vec{a} = 2\hat{i} + \lambda\hat{j} + 1\hat{k}$ $\vec{b} = 1\hat{i} + 2\hat{j} + 3\hat{k}$ Set their dot product to zero for orthogonality: $\vec{a} \cdot \vec{b} = 0$ $(2)(1) + (\lambda)(2) + (1)(3) = 0$ $2 + 2\lambda + 3 = 0$ $5 + 2\lambda = 0$ $2\lambda = -5$ $\lambda = \frac{-5}{2}$

Step 4: Final Answer:
The value of $\lambda$ is $\frac{-5}{2}$.