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For three-dimensional vectors $u, v, w$, the scalar triple product $u\cdot(v\times w)$ is equal to the determinant of the matrix formed by placing these vectors as columns, denoted as $\det[u\ v\ w]$.

Conditions: The vectors $u, v, w$ are three-dimensional.; The vectors are arranged as columns in the matrix.

For three-dimensional vectors $u, v, w$, the scalar triple product $u\cdot(v\times w)$ is equal to the determinant of the matrix formed by placing these vectors as columns, denoted as $\det[u\ v\ w]$.

Conditions: The vectors $u, v, w$ are three-dimensional.; The vectors are arranged as columns in the matrix.

The lesson connects them by showing that the scalar triple product of three vectors equals the determinant of the matrix formed by those vectors as columns. It also clarifies that the cross-product mnemonic uses basis symbols as a formal device, linking the algebraic determinant structure to vector operations.

Conditions: Working in three-dimensional space.; Vectors are treated as columns in the determinant.

The lesson connects them by showing that the scalar triple product of three vectors equals the determinant of the matrix formed by those vectors as columns. It also clarifies that the cross-product mnemonic uses basis symbols as a formal device, linking the algebraic determinant structure to vector operations.

Conditions: Working in three-dimensional space.; Vectors are treated as columns in the determinant.