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Chapter 1: Vectors, Lines and Planes

Section 1.5: Applications of Vector Products

Example 1.5.1

If $A = 3 i - 2 j + 4 j$ , $B = 2 i + 5 j - 4 k$ , and $C = 5 i + 7 j + 6 k$ ,

a)	Compute $[ABC]$ , the Triple Scalar (or Box) Product $A \cdot B \times C$ .

b)	Verify the identity $A \cdot B \times C = A \times B \cdot C$ for the Triple Scalar Product.

Solution

Mathematical Solution

Part (a)

$[ABC]$ = $A \cdot B \times C$ = $| \begin{array}{c} 3 & - 2 & 4 \\ 2 & 5 & - 4 \\ 5 & 7 & 6 \end{array} |$ = $194$

Indeed,

$A \cdot B \times C$ = $[\begin{array}{c} 3 \\ - 2 \\ 4 \end{array}] \cdot [\begin{array}{c} 58 \\ - 32 \\ - 11 \end{array}]$ = $174 + 64 - 44 = 194$

The cross product is computed first, resulting in the dot product between two vectors. If the dot product were done first, producing a scalar, the second operation would be the cross product between that scalar and a vector, an operation not defined. Hence, the notation $A \cdot B \times C$ implies that the cross product must be done first.

Part (b)

$A \times B \cdot C$ = $C \cdot (A \times B)$ = $| \begin{array}{c} 5 & 7 & 6 \\ 3 & - 2 & 4 \\ 2 & 5 & - 4 \end{array} |$ = $- | \begin{array}{c} 3 & - 2 & 4 \\ 5 & 7 & 6 \\ 2 & 5 & - 4 \end{array} |$ = $| \begin{array}{c} 3 & - 2 & 4 \\ 2 & 5 & - 4 \\ 5 & 7 & 6 \end{array} |$ = $A \cdot (B \times C)$

The dot product between two vectors commutes, giving the first equality. The interchange of two adjacent rows in a determinant negates the determinant. This is done twice, first interchanging rows 1 and 2, then interchanging rows 2 and 3. This final determinant now expresses $[ABC]$ = $A \cdot B \times C$ .

Maple Solution - Interactive

Initialization

•	Tools≻Load Package: Student Multivariate Calculus

Loading Student:-MultivariateCalculus

•	Enter A as per Table 1.1.1.

•	Context Panel: Assign to a Name≻A

$⟨3, - 2, 4⟩$ $\overset{assign to a name}{\to}$ $A$

•	Enter B as per Table 1.1.1.

•	Context Panel: Assign to a Name≻B

$⟨2, 5, - 4⟩$ $\overset{assign to a name}{\to}$ $B$

•	Enter C as per Table 1.1.1.

•	Context Panel: Assign to a Name≻C

$⟨5, 7, 6⟩$ $\overset{assign to a name}{\to}$ $C$

Part (a)

Solution via Context Panel

•	Write a sequence of the names of the three vectors. Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Triple Scalar Product

$A, B, C$ = $[\begin{array}{c} 3 \\ −2 \\ 4 \end{array}], [\begin{array}{c} 2 \\ 5 \\ −4 \end{array}], [\begin{array}{c} 5 \\ 7 \\ 6 \end{array}]$ $\overset{scalar triple product}{\to}$ $194$

Solution from first principles

•	Common Symbols palette: Dot-product and cross-product operators Note the (essential) use of parentheses.

•	Context Panel: Evaluate and Display Inline

$A \cdot (B \times C)$ = $194$

Part (b)

Obtain the Box Product $[CAB]$

•	Write a sequence of the names of the three vectors. Context Panel: Evaluate and Display Inline

•	Context Panel: Student Multivariate Calculus≻Triple Scalar Product

$C, A, B$ = $[\begin{array}{c} 5 \\ 7 \\ 6 \end{array}], [\begin{array}{c} 3 \\ −2 \\ 4 \end{array}], [\begin{array}{c} 2 \\ 5 \\ −4 \end{array}]$ $\overset{scalar triple product}{\to}$ $194$

Solution from first principles

•	Common Symbols palette: Dot-product and cross-product operators

•	Context Panel: Evaluate and Display Inline

$A \times B \cdot C$ = $194$

Maple Solution - Coded

Initialization

•	Install the Student MultivariateCalculus package.

$with (Student :- MultivariateCalculus) &colon;$

•	Define the vectors A, B, and C.

$A, B, C ≔ ⟨3, - 2, 4⟩, ⟨2, 5, - 4⟩, ⟨5, 7, 6⟩ &colon;$

Part (a)

•	Apply the BoxProduct command.

$BoxProduct (A, B, C)$ = $194$

Solution from first principles

•	Use the DotProduct and CrossProduct commands to obtain $[ABC]$ = $A \cdot B \times C$ .

$DotProduct (A, CrossProduct (B, C))$ = $194$

In the Student MultivariateCalculus package, the commands BoxProduct and TripleScalarProduct are equivalent. Either form of the command can be used.

Part (b)

Apply the BoxProduct command.

$BoxProduct (A, B, C)$ = $194$

$BoxProduct (C, A, B)$ = $194$

Solution from first principles

•	Use the DotProduct and CrossProduct commands to obtain $A \times B \cdot C$ .

$DotProduct (CrossProduct (A, B), C)$ = $194$

<< Chapter Overview Section 1.5 Next Example >>

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