If vectors a, b, c are coplanar, show...

MathematicsScalar Triple ProductJEE Advanced 1989Moderate

If vectors $a, b, c$ are coplanar, show that $a \cdot a b \cdot a c \cdot a a \cdot b b \cdot b c \cdot b a \cdot c b \cdot c c \cdot c = 0$ .

Visualized Solution (English)

Visualizing Coplanar Vectors

Given vectors $a, b, c$ are coplanar.
This means they all lie in the same two-dimensional plane.

The Condition of Linear Dependence

Since $a, b, c$ are coplanar, they are linearly dependent.
There exist scalars $x, y, z$ (not all zero) such that: $x a + y b + z c = 0$

Dot Product with Vector .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Taking the dot product of the equation with $a$ :
$a \cdot (x a + y b + z c) = a \cdot 0$
Expanding: $x (a \cdot a) + y (a \cdot b) + z (a \cdot c) = 0$

Dot Product with Vector .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b

Taking the dot product of the equation with $b$ :
$b \cdot (x a + y b + z c) = b \cdot 0$
Expanding: $x (b \cdot a) + y (b \cdot b) + z (b \cdot c) = 0$

Dot Product with Vector .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } c

Taking the dot product of the equation with $c$ :
$c \cdot (x a + y b + z c) = c \cdot 0$
Expanding: $x (c \cdot a) + y (c \cdot b) + z (c \cdot c) = 0$

The Homogeneous System

We have a system of three homogeneous equations:
1. $x (a \cdot a) + y (a \cdot b) + z (a \cdot c) = 0$
2. $x (b \cdot a) + y (b \cdot b) + z (b \cdot c) = 0$
3. $x (c \cdot a) + y (c \cdot b) + z (c \cdot c) = 0$

The Determinant Condition

For a non-trivial solution $(x, y, z) \neq = (0, 0, 0)$ , the determinant of the coefficient matrix must be zero:
$a \cdot a b \cdot a c \cdot a a \cdot b b \cdot b c \cdot b a \cdot c b \cdot c c \cdot c = 0$
Hence Proved.

Summary and Beyond

Key Takeaway: The Gram determinant of coplanar vectors is always zero.
Connection: This determinant is equal to $[a b c]^{2}$ .
Next Challenge: What happens to this determinant if the vectors are mutually perpendicular?

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