Vector multiplication can be completed using the dot product or cross product.
The result of the cross product is a vector. We can calculate the cross product using geometric or algebraic vectors.
The algebraic cross product of two vectors \( \vec{a} = \left(a_{1},a_{2},a_{3}\right), \vec{b} = \left(b_{1},b_{2},b_{3}\right) \)
is defined by:
\( \vec{a}\times\vec{b} = \left(a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3}, a_{1}b_{2}-a_{2}b_{1} \right)\)
Given two vectors \( \vec{a} = (2,4,6) \) and \( \vec{b} = (-1,2,-5) \) calculate \( \vec{a}\times\vec{b} \) and \( \vec{b}\times\vec{a} \).
What property does this demonstrate does hold not for the cross product? Explain why the property does not hold.
Show Answer
\( \vec{a}\times\vec{b} = (-20-12, -6+10, 4+4) = (-32, 4, 8) \)
\( \vec{b}\times\vec{a} = (12+20, -10+6, -4-4) = (32,-4,-8) \)
since \( \vec{a}\times\vec{b} \neq \vec{b}\times\vec{a} \) cross product is NOT commutative
Using the two vectors above and a third vector \( \vec{c} = (4,3,-1) \) calculate
1) \(\vec{a}\times(\vec{b}+\vec{c})\) and 2) \( (\vec{a}\times\vec{b}) + (\vec{a}\times\vec{c}) \)
Problem 1) \(\vec{a}\times(\vec{b}+\vec{c})\)
Show Answer
First add the vectors and then perform the cross product:
\(\vec{a}\times(\vec{b}+\vec{c})\) \( = (-24-30, 18+12, 10-12) \)
\( = (-54,30,-2) \)
Problem 2) \( (\vec{a}\times\vec{b}) + (\vec{a}\times\vec{c}) \)
Show Answer
First calculate the cross product then add the vectors:
\( \vec{a}\times\vec{c} = (-4-18, 24+2, 6-16) \)
\( = (-22, -26, -10) \)
\( \therefore (\vec{a}\times\vec{b}) + (\vec{a}\times\vec{c}) \)
\( = (-32, 4, 8) + (-22, -26, -10) \)
\( = (-54,30,-2) \)
Compare the results from above. What property does this demonstrate?
Show Answer
Since the result from i) and ii) is the same - it demonstrates that the cross product is distributive:
\(\vec{a}\times(\vec{b}+\vec{c}) = (\vec{a}\times\vec{b}) + (\vec{a}\times\vec{c})\)
Using the three vectors \( \vec{a}, \vec{b} \text{ and } \vec{c}\) from above calculate
1) \( (\vec{a}\times\vec{b})\times\vec{c}\)
and
2) \(\vec{a}\times(\vec{b}\times\vec{c})\)
Problem 1) \( (\vec{a}\times\vec{b})\times\vec{c}\)
Show Answer
First calculate \( (\vec{a}\times\vec{b}) \):
\( (\vec{a}\times\vec{b})\times\vec{c} = (-32, 4, 8) \times (4,3,-1) \)
\( (\vec{a}\times\vec{b})\times\vec{c} = (-4-24, -32-32, -96-16) \)
\( = (-28,0,-112) \)
Problem 2) \(\vec{a}\times(\vec{b}\times\vec{c})\)
Show Answer
First calculate \( (\vec{b}\times\vec{c}) \):
\( \vec{a}\times(\vec{b}\times\vec{c}) = (2,4,6) \times (13,-21,-11) \)
\( \vec{a}\times(\vec{b}\times\vec{c}) = (-44+126, 78+22, -42-52) \)
\( = (82, 100, -94) \)
What property does this demonstrate does NOT hold for the cross product?
Show Answer
Since the results from 1) and 2) are different the cross product is not associative.