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Scalar Triple Product Calculator

Enter the inputs and click on Calculate button. Empty inputs are considered 0.

\( (\vec{a}\times\vec{b})\cdot\vec{c} = \begin{vmatrix}a_{1} & a_{2} & a_{3}\\b_{1} & b_{2} & b_{3}\\c_{1} & c_{2} & c_{3}\end{vmatrix} \)

= \( \vec{x} \) + \( \vec{y} \) + \( \vec{z} \)
= \( \vec{x} \) + \( \vec{y} \) + \( \vec{z} \)
= \( \vec{x} \) + \( \vec{y} \) + \( \vec{z} \)




What is Scalar Triple Product

The Scalar Triple Product is a way of multiplying three vectors to get a scalar quantity. This product represents the volume of the parallelepiped formed by the three vectors. The result is a single number, which can be positive, negative, or zero, depending on the orientation of the vectors.

Vectors in three-dimensional space can be represented as \(\vec{v} = (v_1, v_2, v_3)\), where \(v_1\), \(v_2\), and \(v_3\) are the components of the vector along the x, y, and z axes, respectively.

Steps to Calculate the Scalar Triple Product

Here are the steps to calculate the scalar triple product of three vectors:

  1. First, calculate the cross product of two of the vectors.
  2. Then, take the dot product of the resulting vector with the third vector.
  3. The resulting value is the scalar triple product.

Formula

The scalar triple product of vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is given by:

\[\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix}a_1 & a_2 & a_3 \\b_1 & b_2 & b_3 \\c_1 & c_2 & c_3 \end{vmatrix}\]

Example

Let's take an example to illustrate the process. Suppose we have three vectors a, b, and c:

\(\vec{a} = (a_1, a_2, a_3)\) = (1, 2, 3)

\(\vec{b} = (b_1, b_2, b_3)\) = (4, 5, 6)

\(\vec{c} = (c_1, c_2, c_3)\) = (7, 8, 9)

To find the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c})\), we follow these steps:

  1. First, calculate the cross product \(\vec{b} \times \vec{c}\):

    2. \( \vec{b} \times \vec{c} = \)\( \left( b_2 c_3 - b_3 c_2, b_3 c_1 - b_1 c_3, b_1 c_2 - b_2 c_1 \right) \)

    Substituting the given values:

    \( \vec{b} \times \vec{c} = \)\( \left( 5 \times 9 - 6 \times 8, 6 \times 7 - 4 \times 9, 4 \times 8 - 5 \times 7 \right) = \)\( \left( 45 - 48, 42 - 36, 32 - 35 \right) = \)\( \left( -3, 6, -3 \right) \)

  2. Next, take the dot product of \(\vec{a}\) with the resulting vector:

    3. \( \vec{a} \cdot (\vec{b} \times \vec{c}) = \)\( a_1 (-3) + a_2 (6) + a_3 (-3) \)

    Substituting the values of \(\vec{a}\):

    \( \vec{a} \cdot (\vec{b} \times \vec{c}) = \)\( 1 \times (-3) + 2 \times 6 + 3 \times (-3) = \)\( -3 + 12 - 9 = 0 \)

The resulting value is 0.

Therefore, the scalar triple product of \(\vec{a} = (1, 2, 3)\), \(\vec{b} = (4, 5, 6)\), and \(\vec{c} = (7, 8, 9)\) is 0.