The scalar triple product (also called the mixed product) of three vectors a, b and c is defined as the dot product of the vector a and the cross product of the other two vectors b and c. As the name suggests, the scalar triple product is a scalar quantity. It is denoted by
If the scalar triple product of vectors a, b and c is zero, then the three vectors are linearly dependent (coplanar), i.e. one of the vectors can be represented as a linear combination of the two other vectors:
A necessary and sufficient condition for three vectors \(\mathbf{a} = \left({X_1, Y_1, Z_1}\right),\) \(\mathbf{b} = \left({X_2, Y_2, Z_2}\right),\) and \(\mathbf{c} = \left({X_3, Y_3, Z_3}\right)\) to be coplanar is that the determinant whose rows are the coordinates of these vectors is equal to zero:
If the scalar triple product of vectors a, b and c is nonzero, then these vectors are linearly independent.
Vector Triple Product
The vector triple product of three vectors a, b and c is defined as the cross product of the vector a with the cross product of the other two vectors b and c.
As the name suggests, the vector triple product is a vector quantity. It can be calculated by the formula
Find the scalar triple product of vectors \(\mathbf{a}\left({1,2,3}\right),\) \(\mathbf{b}\left({1,-1,-1}\right),\) and \(\mathbf{b}\left({1,1,1}\right).\)
Solution.
We can calculate the triple product using the determinant. This yields:
Show that four points \(A\left({-1,2,1}\right),\) \(B\left({0,4,2}\right),\) \(C\left({-1,5,5}\right),\) and \(D\left({2,5,0}\right)\) lie in the same plane.
Solution.
Compute the coordinates of vectors \(\mathbf{a} = \mathbf{AB},\) \(\mathbf{b} = \mathbf{AC},\) and \(\mathbf{c} = \mathbf{AD}:\)
The determinant is zero. Hence, the vectors are coplanar.
Example 3.
Show that vectors \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{k},\) \(\mathbf{b} = \mathbf{i} - \mathbf{j}\) and \(\mathbf{c} = 2\mathbf{j} + 3\mathbf{k}\) are coplanar and express vector \(\mathbf{c}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}.\)
Solution.
The coordinates of the vectors \(\mathbf{a},\) \(\mathbf{b},\) and \(\mathbf{c}\) are equal
Given a pyramid with vertices \(A\left({2,1,0}\right),\) \(B\left({-1,5,0}\right),\) \(C\left({-1,1,0}\right),\) and \(D\left({0,0,6}\right).\) Calculate its volume and the height drawn to the face \(ABC.\)
Solution.
Determine the coordinates of the vectors \(\mathbf{a} = \mathbf{AB},\) \(\mathbf{b} = \mathbf{AC}\) and \(\mathbf{c} = \mathbf{AD}\) that span the pyramid: