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:
We are given three vectors in 3D component form. We need to determine the value of the scalar parameter $\lambda$ such that these three vectors lie entirely within the same flat geometric plane (i.e., they are coplanar).

Step 2: Key Formula or Approach:
For three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ to be coplanar, their scalar triple product must be zero: $$[\vec{a} \quad \vec{b} \quad \vec{c}] = 0$$ This is equivalent to setting the determinant of the matrix formed by their component coefficients equal to zero: $$\begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix} = 0$$

Step 3: Detailed Explanation:
Let's plug the given vector components into the matrix determinant environment: $$\begin{vmatrix} 2 & -1 & -1 \\ 1 & 2 & -3 \\ 3 & \lambda & 5 \end{vmatrix} = 0$$ Expanding the determinant along the first row: $$2((2)(5) - (-3)(\lambda)) - (-1)((1)(5) - (-3)(3)) + (-1)((1)(\lambda) - (2)(3)) = 0$$ Simplify the terms inside each pair of parentheses: $$2(10 + 3\lambda) + 1(5 + 9) - 1(\lambda - 6) = 0$$ $$20 + 6\lambda + 14 - \lambda + 6 = 0$$ Combine the like terms containing $\lambda$ and the constant integers: $$5\lambda + 40 = 0$$ $$5\lambda = -40 \implies \lambda = -8$$

Step 4: Final Answer:
The value of $\lambda$ is $-8$, which matches option (A).