Animated Solution for Mathematics - Vector Algebra: If vectors $\vec{a},\vec{b},\vec{c}$ are coplanar, show that $\left|\begin{array}{ccc}\vec{a}\cdot \vec{a}& \vec{a}\cdot \vec{b}& \vec{a}\cdot \vec{c}\\ \vec{b}\cdot \vec{a}& \vec{b}\cdot \vec{b}& \vec{b}\cdot \vec{c}\\ \vec{c}\cdot \vec{a}& \vec{c}\cdot \vec{b}& \vec{c}\cdot \vec{c}\end{array}\right|=0$.

Unveiling the Gramian Matrix

Imagine you are standing on a perfectly flat, infinite sheet of paper. You have three arrows, $\vec{a}$, $\vec{b}$, and $\vec{c}$, all drawn on this sheet.

Because they are all confined to this two-dimensional world, they are what we call coplanar. In the vast, three-dimensional space of JEE Advanced physics and mathematics, this is a special condition. It means these vectors are not 'free' to explore the third dimension; they are locked in a relationship of linear dependence.

When we say $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar, we are saying that one of them can be written as a combination of the others. Mathematically, this means there exist scalars $x,y,z$, not all of which are zero, such that:

This equation is the heartbeat of our proof. It tells us that these vectors are linearly dependent. If you try to move in the direction of $\vec{a}$, you can always compensate by moving in the directions of $\vec{b}$ and $\vec{c}$ to return to the origin.

To bridge the gap between our linear combination and the matrix, we need a tool that extracts the 'length' and 'angle' information from these vectors. That tool is the dot product.

Let us take our equation $x\vec{a}+y\vec{b}+z\vec{c}=\vec{0}$ and dot it with each vector in turn. First, dotting with $\vec{a}$:

Expanding this, we get:

We repeat this process for $\vec{b}$ and $\vec{c}$. Suddenly, we have a system of three homogeneous linear equations:

Look closely at this system. We have three variables ($x,y,z$) and three equations. We know for a fact that a non-trivial solution exists because the vectors are coplanar.

In the world of linear algebra, a homogeneous system $A\mathbf{v}=0$ has a non-trivial solution if and only if the determinant of the coefficient matrix $A$ is zero. Constructing the matrix from our coefficients, we arrive at the final, elegant result:

This matrix is known as the Gramian matrix. Its determinant represents the square of the volume of the parallelepiped formed by the vectors.

Since our vectors are coplanar, they enclose zero volume in 3D space, and thus, the determinant must be zero. You have just proven a fundamental property of vector geometry using the power of linear algebra. Take a moment to appreciate the symmetry and the logic—this is the beauty of JEE mathematics!