Definition:Scalar Triple Product/Definition 2
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Definition
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.
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.
Also known as
The scalar triple product is also known as the triple scalar product.
Some sources denote the scalar triple product as $\sqbrk {\mathbf a \mathbf b \mathbf c}$.
Also see
- Results about scalar triple product can be found here.
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
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 4$