Question

If the vectors \[ \vec a=4\hat{i}+6\hat{j}+\lambda\hat{k},\qquad \vec b=\hat{i}-2\hat{j}-3\hat{k}, \qquad \vec c=4\lambda\hat{i}+\hat{j}-3\hat{k} \] are coplanar and \(\lambda\in\mathbb{Z}\), then \(\vec a\cdot\vec c=\)

• A. \(3\)
• B. \(13\)
• C. \(25\)
• D. \(32\)

Hint:

Three vectors are coplanar if their scalar triple product is zero: \[ \boxed{ \left| \begin{array}{ccc} a_1& a_2& a_3 b_1& b_2& b_3 c_1& c_2& c_3 \end{array} \right|=0. } \] After finding the unknown parameter, substitute it to evaluate the required expression.

Answer 1

The Correct Option is D