Equivalence of Definitions of Scalar Triple Product
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Theorem
The following definitions of the concept of Scalar Triple Product are equivalent:
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:
\(\ds \mathbf a\) | \(=\) | \(\ds a_i \mathbf i + a_j \mathbf j + a_k \mathbf k\) | ||||||||||||
\(\ds \mathbf b\) | \(=\) | \(\ds b_i \mathbf i + b_j \mathbf j + b_k \mathbf k\) | ||||||||||||
\(\ds \mathbf c\) | \(=\) | \(\ds c_i \mathbf i + c_j \mathbf j + c_k \mathbf k\) |
where $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis.
Definition 1
The scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$ is defined and denoted as:
- $\sqbrk {\mathbf a, \mathbf b, \mathbf c} := \mathbf a \cdot \paren {\mathbf b \times \mathbf c}$
where:
- $\cdot$ denotes dot product
- $\times$ denotes vector cross product.
Definition 2
The scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$ is defined and denoted as:
- $\sqbrk {\mathbf a, \mathbf b, \mathbf c} := \begin {vmatrix}
a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \\ \end {vmatrix}$
where $\begin {vmatrix} \ldots \end {vmatrix}$ is interpreted as a determinant.
Proof
\(\ds \mathbf a \cdot \paren {\mathbf b \times \mathbf c}\) | \(=\) | \(\ds \mathbf a \cdot \paren {\paren {b_j c_k - c_j b_k} \mathbf i + \paren {b_k c_i - c_k b_i} \mathbf j + \paren {b_i c_j - c_i b_j} \mathbf k}\) | Definition of Vector Cross Product | |||||||||||
\(\ds \) | \(=\) | \(\ds a_i \paren {b_j c_k - c_j b_k} + a_j \paren {b_k c_i - c_k b_i} + a_k \paren {b_i c_j - c_i b_j}\) | Definition of Dot Product |
Then:
\(\ds \begin {vmatrix} a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \end {vmatrix}\) | \(=\) | \(\ds a_i b_j c_k - a_i b_k c_j - a_j b_i c_k + a_j b_k c_i + a_k b_i c_j - a_k b_j c_i\) | Determinant of Order 3 | |||||||||||
\(\ds \) | \(=\) | \(\ds a_i \paren {b_j c_k - c_j b_k} + a_j \paren {b_k c_i - c_k b_i} + a_k \paren {b_i c_j - c_i b_j}\) | extracting factors |
Hence the result.
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $7$. Products of Three Vectors
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Miscellaneous Formulas involving Dot and Cross Products: $22.16$