Definition

Scalar Triple Product

$a \cdot (b \times c) = b \cdot (c \times a) = c \cdot (a \times b)$ The signed volume of the parallelepiped defined by the three vectors.

Vector Triple Product

$a \times (b \times c) = b (a \cdot c) - c (a \cdot b)$

Facts

$a \cdot (b \times c) = det [a, b, c]$

proof

$(u \times v) \cdot w = (u_{2} v_{3} - u_{3} v_{2}, - (u_{1} v_{3} - u_{3} v_{1}), u_{1} v_{2} - u_{2} v_{1}) \cdot w = [w_{1} (u_{2} v_{3} - u_{3} v_{2}, - w_{2} (u_{1} v_{3} - u_{3} v_{1}), w_{3} (u_{1} v_{2} - u_{2} v_{1})]$ $= w_{1} u_{2} v_{2} u_{3} v_{3} - w_{2} u_{1} v_{1} u_{3} v_{3} + w_{3} u_{1} v_{1} u_{2} v_{2} = w_{1} u_{1} v_{1} w_{2} u_{2} v_{2} w_{3} u_{3} v_{3} = det [u, v, w]$

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Triple Product

Definition

Scalar Triple Product

Vector Triple Product

Facts

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