Animated Solution for Mathematics - Vector Algebra: For three vectors $\vec{u},\vec{v},\vec{w}$ which of the following expression is not equal to any of the remaining three?

The Scalar Triple Product (STP), denoted as $\vec{u}\cdot (\vec{v}\times \vec{w})$, is the mathematical engine that calculates the volume of a parallelepiped defined by vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$.

The cross product $\vec{v}\times \vec{w}$ generates a vector whose magnitude equals the area of the base parallelogram and whose direction is perpendicular to that base.

When we compute the dot product of $\vec{u}$ with this area vector, we project $\vec{u}$ onto the perpendicular direction to determine the height. Since volume is defined as base area multiplied by height, the expression $\vec{u}\cdot (\vec{v}\times \vec{w})$ yields the volume.

The Cyclic Property

If we rotate the vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$ while maintaining their cyclic order, the volume remains invariant. This is known as the Cyclic Property:

This property allows you to rearrange vectors within the triple product to simplify complex calculations without altering the physical volume. It essentially preserves the 'right-handed' orientation of the system.

When the Carousel Breaks

If the cyclic order is disrupted—for instance, by swapping two vectors—the orientation of the system is reversed. This transformation turns a right-handed system into a left-handed one.

Mathematically, this swap introduces a negative sign:

This 'Anti-Cyclic' penalty is a common pitfall. Always verify the sequence of vectors; if the order is not cyclic, you must account for the sign change.

The Final Verdict

We evaluate the four given expressions to identify the one that does not represent the standard volume $[\vec{u}\vec{v}\vec{w}]$:

1. Option (a): $\vec{u}\cdot (\vec{v}\times \vec{w})$ is the standard definition, $[\vec{u}\vec{v}\vec{w}]$.

2. Option (b): $(\vec{v}\times \vec{w})\cdot \vec{u}$ is identical to Option (a) due to the commutative property of the dot product.

3. Option (d): $(\vec{u}\times \vec{v})\cdot \vec{w}$ preserves the cyclic order $\vec{u}\to \vec{v}\to \vec{w}$, and is therefore equal to $[\vec{u}\vec{v}\vec{w}]$.

4. Option (c): $\vec{v}\cdot (\vec{u}\times \vec{w})$ follows the order $\vec{v},\vec{u},\vec{w}$. This is an anti-cyclic permutation, resulting in $-[\vec{u}\vec{v}\vec{w}]$.

Conclusion: Option (c) is the odd one out, as it represents the negative of the volume defined by the other three expressions.