Question

The sum of all real values of \(λ\) for which the vectors \(\overset{⃗}{a} = λ\hat{i}+\hat{j}+\hat{k}\), \(\overset{⃗}{b} = \hat{i}+λ\hat{j}+2\hat{k}\), \(\overset{⃗}{c} = 2\hat{i}+3\hat{j}+λ\hat{k}\) are coplanar is...

• A. \(9\)
• B. \(7\)
• C. \(0\)
• D. cant determine

Hint:

Coplanar vectors have a zero scalar triple product; sum the real roots of the cubic.

Answer 1

The Correct Option is C

Step 1: Condition:
Three vectors are coplanar when \(\begin{vmatrix}\lambda&1&1\\1&\lambda&2\\2&3&\lambda\end{vmatrix}=0\).

Step 2: Expand:
\[ \lambda(\lambda^2-6)-1(\lambda-4)+1(3-2\lambda)=\lambda^3-6\lambda-\lambda+4+3-2\lambda=\lambda^3-9\lambda+7 \]

Step 3: Roots:
We need the sum of real roots of \(\lambda^3-9\lambda+7=0\). The cubic has values \(f(-4)=-21<0\), \(f(0)=7>0\), \(f(2)=-3<0\), \(f(3)=7>0\), so it changes sign three times and has three real roots.

Step 4: Sum:
For \(\lambda^3+0\lambda^2-9\lambda+7=0\), the sum of roots is \(-\) (coefficient of \(\lambda^2\)) \(=0\). Option (C).

Final Answer:
The sum of the values of \(\lambda\) is 0, option (C). \[ \boxed{\text{(C) } 0} \]