Question

If the vectors $2\hat{i} - \hat{j} - \hat{k}$, $\hat{i} + 2\hat{j} - 3\hat{k}$ and $3\hat{i} + \lambda\hat{j} + 5\hat{k}$ are coplanar, then the value of $\lambda$ is

• A. $-8$
• B. $-4$
• C. $-2$
• D. $-1$

Hint:

To perform rapid determinant expansion in coplanar vector problems, look for rows containing small integers like 1 to expand along, or apply simple row operations (such as $R_1 \rightarrow R_1 - 2R_2$) to create zeros before expanding!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Three vectors are given; we must find λ such that they are coplanar.

Step 2: Key Formula or Approach:
Vectors are coplanar if their scalar triple product (determinant) equals zero.

Step 3: Detailed Explanation:
Setting the determinant of [2,-1,-1; 1,2,-3; 3,λ,5] to zero: 2(10+3λ) + 1(5+9) – 1(λ–6) = 0 → 20+6λ+14–λ+6 = 0 → 5λ+40 = 0 → λ = –8.

Step 4: Final Answer:
The value is λ = –8, matching option (A).