Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
$\vec{a}$ and $\vec{b}$ are two non-collinear vectors then the points with position vectors $l_1\vec{a} + m_1\vec{b}, l_2\vec{a} + m_2\vec{b}, l_3\vec{a} + m_3\vec{b}$ are collinear then find the value of $\begin{vmatrix} 1 & 1 & 1 \\ l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{vmatrix}$.
Step-by-Step Solution
Key Concept: Colinearity of three position vectors requires the cross product of two difference vectors to be zero, which gives a determinant condition.
Given $\vec{r_1} = l_1\vec{a} + m_1\vec{b}$, $\vec{r_2} = l_2\vec{a} + m_2\vec{b}$, $\vec{r_3} = l_3\vec{a} + m_3\vec{b}$ are colinear, we have $(\vec{r_1} - \vec{r_2}) \times (\vec{r_2} - \vec{r_3}) = 0$. This simplifies to the determinant condition $(l_1 - l_2)(m_2 - m_3) - (m_1 - m_2)(l_2 - l_3) = 0$.
Correct Answer: 0