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scalar triple product


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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:(i) [a, b, c] =−[a, c, b].(ii)[a, b, c]=[b, c, a]=[c, a, b].(iii) The vectors a, b and c are coplanar if and only if [a, b, c]=0(iv) If the vectors are given in terms of their components (with respect to the standard vectors i, j and k) as a = a1i + a2j + a3k, b = b1i + b2j + b3k, and c = c1i + c2j + c3k, then(v) Let, and represent 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.)

(i) [a, b, c] =−[a, c, b].

(ii)[a, b, c]=[b, c, a]=[c, a, b].

(iii) The vectors a, b and c are coplanar if and only if [a, b, c]=0

(iv) If the vectors are given in terms of their components (with respect to the standard vectors i, j and k) as a = a1i + a2j + a3k, b = b1i + b2j + b3k, and c = c1i + c2j + c3k, then

(v) Let, and represent 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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