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3-3 (l) The scalar triple product

3-3 (l) The scalar triple product

Point 6 of 7 — The Vector (Cross) Product and the Scalar Triple ProductQuestion bank: 0 questions (0 with pictures)

06 3-3 (l) The scalar triple product

Definition

  • For three vectors A⃗, B⃗, C⃗ the scalar triple product is A⃗⋅(B⃗×C⃗)
  • The brackets are essential in the sense that B⃗×C⃗ is calculated first (the other order, (A⃗⋅B⃗)×C⃗, is meaningless since A⃗⋅B⃗ is a number).
  • The result is a number.

Determinant form

  • If A⃗=(Ax,Ay,Az), B⃗=(Bx,By,Bz), C⃗=(Cx,Cy,Cz), then A⃗⋅(B⃗×C⃗)=∣AxAyAzBxByBzCxCyCz∣ =Ax(ByCz−BzCy)−Ay(BxCz−BzCx)+Az(BxCy−ByCx)

Properties

  • Cyclic permutation keeps the value: A⃗⋅(B⃗×C⃗)=B⃗⋅(C⃗×A⃗)=C⃗⋅(A⃗×B⃗).
  • Interchanging two vectors changes the sign (two rows of the determinant swap).
  • A⃗⋅(B⃗×C⃗)=(A⃗×B⃗)⋅C⃗, so the dot and the cross may be exchanged.
  • If two of the vectors are equal or parallel, the product is zero.
  • If the three vectors lie in one plane (coplanar), the product is zero, and conversely.
✍️ Worked examples

Example 13: Calculate A⃗⋅(B⃗×C⃗) for A⃗=(2,1,3), B⃗=(−1,3,2), C⃗=(1,1,−2).

Solution:

  • Determinant: 2[3(−2)−2(1)]−1[(−1)(−2)−2(1)]+3[(−1)(1)−3(1)].
  • =2(−8)−1(0)+3(−4)=−16−0−12=−28.

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