Triple Product Calculator

The calculator will calculate the triple product (both scalar and vector) of the three vectors, with steps shown.

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Your Input

Calculate $$$\left(-2, 3, 1\right)\cdot \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right)$$$, $$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\cdot \left(-3, 2, 1\right)$$$, $$$\left(-2, 3, 1\right)\times \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right)$$$, and $$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\times \left(-3, 2, 1\right)$$$.

Solution

  • Calculate the scalar triple product $$$\left(-2, 3, 1\right)\cdot \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right)$$$.

    $$$\left(-2, 3, 1\right)\cdot \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right) = \left(-2, 3, 1\right)\cdot \left(-4, -7, 2\right)$$$ (for steps, see cross product calculator).

    Next, $$$\left(-2, 3, 1\right)\cdot \left(-4, -7, 2\right) = -11$$$ (for steps, see dot product calculator).

    The scalar triple product can be found as the determinant that has three vectors as its rows or columns.

  • Calculate the scalar triple product $$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\cdot \left(-3, 2, 1\right)$$$.

    $$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\cdot \left(-3, 2, 1\right) = \left(4, 7, -13\right)\cdot \left(-3, 2, 1\right)$$$ (for steps, see cross product calculator).

    Next, $$$\left(4, 7, -13\right)\cdot \left(-3, 2, 1\right) = -11$$$ (for steps, see dot product calculator).

    The scalar triple product can be found as the determinant that has three vectors as its rows or columns.

  • Calculate the vector triple product $$$\left(-2, 3, 1\right)\times \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right)$$$.

    $$$\left(-2, 3, 1\right)\times \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right) = \left(-2, 3, 1\right)\times \left(-4, -7, 2\right)$$$ (for steps, see cross product calculator).

    Next, $$$\left(-2, 3, 1\right)\times \left(-4, -7, 2\right) = \left(13, 0, 26\right)$$$ (for steps, see cross product calculator).

  • Calculate the vector triple product $$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\times \left(-3, 2, 1\right)$$$.

    $$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\times \left(-3, 2, 1\right) = \left(4, 7, -13\right)\times \left(-3, 2, 1\right)$$$ (for steps, see cross product calculator).

    Next, $$$\left(4, 7, -13\right)\times \left(-3, 2, 1\right) = \left(33, 35, 29\right)$$$ (for steps, see cross product calculator).

Answer

$$$\left(-2, 3, 1\right)\cdot \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right) = -11$$$A

$$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\cdot \left(-3, 2, 1\right) = -11$$$A

$$$\left(-2, 3, 1\right)\times \left(\left(7, -4, 0\right)\times \left(-3, 2, 1\right)\right) = \left(13, 0, 26\right)$$$A

$$$\left(\left(-2, 3, 1\right)\times \left(7, -4, 0\right)\right)\times \left(-3, 2, 1\right) = \left(33, 35, 29\right)$$$A