In this article, we will discuss how to compute the scalar triple product of vectors. But before proceeding further, let us define a vector first.

A vector is a quantity that is depicted by magnitude, as well as direction. We represent vectors with alphabets with a right head arrow on the top that depicts their direction. For example, \overrightarrow {s}, \overrightarrow {t}, and \overrightarrow {u} are representations of vectors.

Scalar Triple Product of Vectors

Scalar triple product is also known as a mixed product. The scalar triple product of three vectors \overrightarrow {u}, \overrightarrow {v}, and \overrightarrow {w} is mathematically denoted as [\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] and it is equal to the dot product of the first vector \overrrightarrow {u} by the cross product of other two vectors \overrightarrow {v} and \overrightarrow {w}. It is called a scalar product because similar to a dot product, the scalar triple product yields a single number.

We can denote this product mathematically as:

[\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = \overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w})

The cross product of these vectors is equal to the determinant. The rows of this determinant are equal to the coordinates of the vectors regarding an orthonormal basis.

[\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] =\begin {bmatrix} \overrightarrow {u_1} & \overrightarrow {u_2} & \overrightarrow {u_3} \\ \overrightarrow {v_1} & \overrightarrow {v_2} & \overrightarrow {v_3} \\ \overrightarrow {w_1} & \overrightarrow {w_2} & \overrightarrow {w_3} \\ \end {bmatrix}

Triple Product Properties

The properties of triple product are given below:

  • If the order of the factors is circularly rotated, then the triple product remains unaffected. For instance, [\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = [\overrightarrow {v}, \overrightarrow {w}, \overrightarrow {u}] = [\overrightarrow {w}, \overrightarrow {u}, \overrightarrow {v}]
  • However, the triple product changes the signs, if the factors are transposed. For instance, [\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = - [\overrightarrow {v}, \overrightarrow {u}, \overrightarrow {w}] = - [\overrightarrow {u}, \overrightarrow {w}, \overrightarrow {v}] = - [\overrightarrow {w}, \overrightarrow {v}, \overrightarrow {u}]
  • If three vectors are linearly dependent to each other, then the triple product is equal to zero.
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Example 1

Find the scalar triple product of the following vectors:

\overrightarrow {u} = (2, -1, 3)     \overrightarrow {v} = (0, 2, -5)     \overrightarrow {w} = (1, -1, -2)

Solution

Mathematically a scalar triple product is represented as:

[\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = \overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w})

First, we will compute the product of \overrightarrow {v} \times \overrightarrow {w} by using a determinant. The coordinates of these vectors will be the elements of the determinant.

\overrightarrow {v} \times \overrightarrow {w} = \begin {vmatrix} \overrightarrow {i} & \overrightarrow {j} & \overrightarrow {k} \\ 0 & 2 & -5\\ 1 & -1 & -2 \\ \end {bmatrix}

We will calculate the determinant using the formula of finding the determinant of a 3x3 matrix like this:

|A| = \overrightarrow{i} \cdot\begin {vmatrix} 2 & -5 \\ -1 & -2\end {vmatrix} - \overrightarrow{j} \cdot \begin {vmatrix} 0 & -5 \\ 1 & -2\end {vmatrix} + \overrightarrow {k} \begin {vmatrix} 0 & 2 \\ 1 & -1\end {vmatrix}

= 9 \overrightarrow {i} - 5 \overrightarrow {j} - 2 \overrightarrow {k}

Now, we will calculate the dot product of \overrightarrow {u} and \overrightarrow{v} \times \overrightarrow {w} like this:

\overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w}) = (2, -1, 3) \cdot (-9, -5, -2) = -18 + 5 - 6 = -19

 

Example 2

Find the scalar triple product of the following vectors:

\overrightarrow {u} = (1, 1, 2)     \overrightarrow {v} = (1, 2, -1)     \overrightarrow {w} = (0, 3, 4)

Solution

Mathematically a scalar triple product is represented as:

[\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = \overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w})

First, we will compute the product of \overrightarrow {v} \times \overrightarrow {w} by using determinant. The coordinates of these vectors will be the elements of the determinant.

\overrightarrow {v} \times \overrightarrow {w} = \begin {vmatrix} \overrightarrow {i} & \overrightarrow {j} & \overrightarrow {k} \\ 1 & 2 & -1\\ 0 & 3 & 4 \\ \end {bmatrix}

We will calculate the determinant using the formula of finding the determinant of a 3x3 matrix like this:

|A| = \overrightarrow{i} \cdot\begin {vmatrix} 2 & -1 \\ 3 & 4\end {vmatrix} - \overrightarrow{j} \cdot \begin {vmatrix} 1 & -1 \\ 0 & 4\end {vmatrix} + \overrightarrow {k} \begin {vmatrix} 1 & 2 \\ 0 & 3\end {vmatrix}

= 11 \overrightarrow {i} - 4 \overrightarrow {j} + 3\overrightarrow {k}

Now, we will calculate the dot product of \overrightarrow {u} and \overrightarrow{v} \times \overrightarrow {w} like this:

\overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w}) = (1, 1, 2) \cdot (11, -4, 3) = 11 - 4 + 6 = 13

 

Example 3

Find the scalar triple product of the following vectors:

\overrightarrow {u} = (0, 2, 3)     \overrightarrow {v} = (4, 3, -1)     \overrightarrow {w} = (0, 1, 5)

Solution

Mathematically a scalar triple product is represented as:

[\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = \overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w})

First, we will compute the product of \overrightarrow {v} \times \overrightarrow {w} by using determinant. The coordinates of these vectors will be the elements of the determinant.

\overrightarrow {v} \times \overrightarrow {w} = \begin {vmatrix} \overrightarrow {i} & \overrightarrow {j} & \overrightarrow {k} \\ 4 & 3 & -1\\ 0 & 1 & 5 \\ \end {bmatrix}

We will calculate the determinant using the formula of finding the determinant of a 3x3 matrix like this:

|A| = \overrightarrow{i} \cdot\begin {vmatrix} 3 & -1 \\ 1 & 5\end {vmatrix} - \overrightarrow{j} \cdot \begin {vmatrix} 4 & -1 \\ 0 & 5\end {vmatrix} + \overrightarrow {k} \begin {vmatrix} 4 & 3 \\ 0 & 1\end {vmatrix}

= 16 \overrightarrow {i} - 20 \overrightarrow {j} + 4\overrightarrow {k}

Now, we will calculate the dot product of \overrightarrow {u} and \overrightarrow{v} \times \overrightarrow {w} like this:

\overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w}) = (0, 2, 3) \cdot (16, -20, 4) = 0 - 40 + 12 = -28

 

Example 4

Find the scalar triple product of the following vectors:

\overrightarrow {u} = (1, 4, 7)     \overrightarrow {v} = (2, -1, -3)     \overrightarrow {w} = (0, 4, 5)

Solution

Mathematically a scalar triple product is represented as:

[\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = \overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w})

First, we will compute the product of \overrightarrow {v} \times \overrightarrow {w} by using determinant. The coordinates of these vectors will be the elements of the determinant.

\overrightarrow {v} \times \overrightarrow {w} = \begin {vmatrix} \overrightarrow {i} & \overrightarrow {j} & \overrightarrow {k} \\ 2 & -1 & -3\\ 0 & 4 & 5 \\ \end {bmatrix}

We will calculate the determinant using the formula of finding the determinant of a 3x3 matrix like this:

|A| = \overrightarrow{i} \cdot\begin {vmatrix} -1 & -3 \\ 4 & 5\end {vmatrix} - \overrightarrow{j} \cdot \begin {vmatrix} 2 & -3 \\ 0 & 5\end {vmatrix} + \overrightarrow {k} \begin {vmatrix} 2 & -1 \\ 0 & 4\end {vmatrix}

= 7 \overrightarrow {i} - 10 \overrightarrow {j} + 8\overrightarrow {k}

Now, we will calculate the dot product of \overrightarrow {u} and \overrightarrow{v} \times \overrightarrow {w} like this:

\overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w}) = (1, 4, 7) \cdot (7, -10, 8) = 7 - 40 + 56 = 23

Example 5

Find the scalar triple product of the following vectors:

\overrightarrow {u} = (4, 1, 0)     \overrightarrow {v} = (6, 2, 7)     \overrightarrow {w} = (1, -4, 2)

Solution

Mathematically a scalar triple product is represented as:

[\overrightarrow {u}, \overrightarrow {v}, \overrightarrow {w}] = \overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w})

First, we will compute the product of \overrightarrow {v} \times \overrightarrow {w} by using determinant. The coordinates of these vectors will be the elements of the determinant.

\overrightarrow {v} \times \overrightarrow {w} = \begin {vmatrix} \overrightarrow {i} & \overrightarrow {j} & \overrightarrow {k} \\ 6 & 2 & 7\\ 1 & -4 & 2 \\ \end {bmatrix}

We will calculate the determinant using the formula of finding the determinant of a 3x3 matrix like this:

= \overrightarrow{i} \cdot\begin {vmatrix} 2 & 7 \\ -4 & 2\end {vmatrix} - \overrightarrow{j} \cdot \begin {vmatrix} 6 & 7 \\ 1 & 2\end {vmatrix} + \overrightarrow {k} \begin {vmatrix} 6 & 2\\ 1 & -4\end {vmatrix}

= 32 \overrightarrow {i} - 5 \overrightarrow {j} - 26\overrightarrow {k}

Now, we will calculate the dot product of \overrightarrow {u} and \overrightarrow{v} \times \overrightarrow {w} like this:

\overrightarrow {u} \cdot (\overrightarrow {v} \times \overrightarrow {w}) = (4, 1, 0) \cdot (32, -5, -26) = 128 - 5 - 0 = 123

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Rafia Shabbir