Question

Find the value of $\lambda$ if four points with position vectors $3i+6j+9k,i+2j+3k,2i+3j+k$ and $4i+6j+\lambda k$ are coplanar.

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is A

$2$

Step 1: Find the position vectors.

We have been given four points with position vectors$3i+6j+9k,i+2j+3k,2i+3j+k$ and $4i+6j+\lambda k$ are coplanar.

We need to find the value of$\lambda$.

Let the points with position vectors are,

$$\begin{gathered} \Rightarrow OA=3i+6j+9k, \\ \Rightarrow OB=i+2j+3k, \\ \Rightarrow OC=2i+3j+k, \\ \Rightarrow OD=4i+6j+\lambda k \end{gathered}$$

So, the vector $AB$ would be,

$$\begin{gathered} \Rightarrow AB=OB–OA \\ \Rightarrow AB=(1-3)i+(2-6)j+(3-9)k \\ \Rightarrow AB=-2i-4j-6k \end{gathered}$$

Step 2: Similarly find vectors $AC$and $AD$

$$\begin{gathered} \Rightarrow AC=OC-OA \\ \Rightarrow AC=(2-3)i+(3-6)j+(1-9)k \\ \Rightarrow AC=-i+(-3)j-8k \end{gathered}$$

and

$$\begin{gathered} \Rightarrow AD=OD–OA \\ \Rightarrow AD=(4-3)i+(6-6)j+(\lambda -9)k \\ \Rightarrow AD=i+0j+(\lambda -9)k \end{gathered}$$

Step 3: Find the scalar triple product of vectors $AB,AC$and $AD$

$$\begin{gathered} \Rightarrow [ABACAD]=\left|\begin{array}{ccc}-2& -4& -6\\ -1& -3& -8\\ 1& 0& \lambda -9\end{array}\right| \\ \Rightarrow [ABACAD]=-2\left[\right(-3)\times (\lambda -9)-0]-(-4)\left[\right(-1)\times (\lambda -9)-1\times (-8\left)\right]+(-6)[0-1\times (-3\left)\right] \\ \Rightarrow [ABACAD]=-2(-3\lambda +27)+4(-\lambda +9+8)-6\left(3\right) \\ \Rightarrow [ABACAD]=6\lambda -54-4\lambda +68-18 \\ \Rightarrow [ABACAD]=2\lambda -4 \end{gathered}$$

Step 4: Find the value of $\lambda$by equating the value of the scalar triple product to zero.

As vectors $AB,AC$and $AD$ are coplanar, the scalar triple product is zero.

$$\begin{gathered} \Rightarrow [ABACAD]=0 \\ \Rightarrow 2\lambda -4=0 \\ \Rightarrow \mathbf{}\mathit{\lambda }\mathbf{=}\mathbf{2} \end{gathered}$$

Therefore, option (A) is the correct answer.