Product of Three Vectors

Excerpts

Product of Three Vectors
The third product $\mathrm{S}(\mathrm{V}AB)C$ is represented by the volume of the parallelepiped formed by the vectors $A, B, C$ taken in that order.
Product of Three Vectors
Hence the volume formed by the three vectors has no direction in space, but it is positive or negative according to the cyclical order of the vectors.
Product of Three Vectors
It is merely the third vector multiplied by the scalar product of the other two, or weighted by that product as an ordinary algebraic quantity.
Product of Three Vectors
Hence any of the three former may be expressed by $\mathrm{S}ABC$, and any of the three latter by $-\mathrm{S}ABC$.
Product of Three Vectors
The principle here proved is of great use in solving equations (see p. 455).

Equations

Product of Three Vectors
\mathrm{V}(\mathrm{V}AB)C = -\mathrm{S}BC \cdot A + \mathrm{S}CA \cdot B
The vector product of the vector product of A and B with C equals SCA times B minus SBC times A.
Product of Three Vectors
\mathrm{V}(ABC) = \mathrm{S}AB \cdot C - \mathrm{S}BC \cdot A + \mathrm{S}CA \cdot B
The total vector product of A, B, C is the sum of the first and second partial products, written as scalar products times vectors.
Product of Three Vectors
\mathrm{V}(\alpha\beta\gamma) &= \cos \alpha\beta \cdot \gamma - \cos \beta\gamma \cdot \alpha + \cos \gamma\alpha \cdot \beta
For unit vectors alpha, beta, gamma, the total vector product equals cos(alpha beta) times gamma minus cos(beta gamma) times alpha plus cos(gamma alpha) times beta.
Product of Three Vectors
\mathrm{V}(\beta\gamma\alpha) &= \cos \beta\gamma \cdot \alpha - \cos \gamma \alpha \cdot \beta + \cos \alpha \beta \cdot\gamma
By cyclic permutation of alpha, beta, gamma, the total vector product of beta, gamma, alpha has the analogous form.
Product of Three Vectors
\mathrm{V}(\gamma\alpha\beta) &= \cos \gamma\alpha \cdot \beta - \cos \alpha\beta \cdot \gamma + \cos \beta\gamma \cdot \alpha
By cyclic permutation again, the total vector product of gamma, alpha, beta has the analogous form.

Problems

Exercise Probs-38-43

Exercise Probs-38-43, problem 38
Find the second partial product of $9\, \overline{20^\circ/}\!\underline{/30^\circ}$, $10\, \overline{30^\circ/}\!\underline{/40^\circ}$, $11\, \overline{45^\circ/}\!\underline{/45^\circ}$. Also the third partial product.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Probs-38-43, problem 39
Find the cosine of the angle between the plane of $l_1 i + m_1 j + n_1 k$ and $l_2 i + m_2 j + n_2 k$ and the plane of $l_3 i + m_3 j + n_3 k$ and $l_4 i + m_4 j + n_4 k$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise Probs-38-43, problem 40
Find the volume of the parallelepiped determined by the vectors $100i + 50j + 25k$, $50i + 10j + 80k$, and $-75i + 40j - 80k$.
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- (none printed)
unverified: no computed check settled this one (yet)
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- evaluate: no printed answer to check
Exercise Probs-38-43, problem 41
Find the volume of the tetrahedron determined by the extremities of the following vectors: $3i - 2j + 1k$, $-4i + 5j - 7k$, $3i - 7j - 2k$, $8i + 4j - 3k$.
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- (none printed)
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Exercise Probs-38-43, problem 42
Find the voltage at the terminals of a conductor when its velocity is 1500 centimeters per second, the intensity of the magnetic flux is 7000 lines per square centimeter, and the length of the conductor is 20 centimeters, the angle between the first and second being $30^\circ$, and that between the plane of the first two and the direction of the third $60^\circ$.
Printed answer:
- $.91$ volts
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(91, 100)
On the STU-32 (STU, rpn):
```
7000 ENTER 20 ×
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1500 ×
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```
1 E 8 ÷
```
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30 SIN ×
```
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60 SIN ×
```
Calculator: +9093266739736605791019093292905830E-34; the book prints 0.91. Run on the calculator core at firmware 628c96c.
Exercise Probs-38-43, problem 43
Let $\alpha = \overline{20^\circ/}\! \underline{/10^\circ}$, $\beta = \overline{30^\circ/}\! \underline{/25^\circ}$, $\gamma=\overline{40^\circ/}\! \underline{/35^\circ}$. Find $\mathrm{V}\alpha\beta\gamma$, and deduce $\mathrm{V}\beta\gamma\alpha$ and $\mathrm{V}\gamma\alpha\beta$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles