## Quick Reference

For vectors **a**, **b** and **c**, the scalar product, **a**. (**b**×**c**), of **a** with the vector **b**×**c** (see vector product), is called a scalar triple product. It is a scalar quantity and is denoted by [**a**, **b**, **c**]. It has the following properties:**a**, **b**, **c**] =−[**a**, **c**, **b**].**[a**, **b**, **c]=[b**, **c**, **a]=[c**, **a**, **b]**.**a**, **b** and **c** are coplanar if and only if [**a**, **b**, **c**]=0**i**, **j** and **k**) as **a** = *a*_{1}**i** + *a*_{2}**j** + *a*_{3}**k**, **b** = *b*_{1}**i** + *b*_{2}**j** + *b*_{3}**k**, and **c** = *c*_{1}**i** + *c*_{2}**j** + *c*_{3}**k**, then**a**, **b** and **c** and let *P* be the parallelepiped with *OA*, *OB* and *OC* as three of its edges. The absolute value of [**a**, **b**, **c**] then gives the volume of the parallelepiped *P*. (If **a**, **b** and **c** form a right-handed system, then [**a**, **b**, **c**] is positive; and if **a**, **b** and **c** form a left-handed system, then [**a**, **b**, **c**] is negative.)

**a**, **b**, **c**] =−[**a**, **c**, **b**].

**[a**, **b**, **c]=[b**, **c**, **a]=[c**, **a**, **b]**.

**a**, **b** and **c** are coplanar if and only if [**a**, **b**, **c**]=0

**i**, **j** and **k**) as **a** = *a*_{1}**i** + *a*_{2}**j** + *a*_{3}**k**, **b** = *b*_{1}**i** + *b*_{2}**j** + *b*_{3}**k**, and **c** = *c*_{1}**i** + *c*_{2}**j** + *c*_{3}**k**, then

**a**, **b** and **c** and let *P* be the parallelepiped with *OA*, *OB* and *OC* as three of its edges. The absolute value of [**a**, **b**, **c**] then gives the volume of the parallelepiped *P*. (If **a**, **b** and **c** form a right-handed system, then [**a**, **b**, **c**] is positive; and if **a**, **b** and **c** form a left-handed system, then [**a**, **b**, **c**] is negative.)

*Subjects:*
Mathematics.

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