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 geometric cross product of two vectors \( \vec{a}\) and \(\vec{b}\) is defined by:
\( \vec{a}\times\vec{b} = \vert\vec{a}\vert\vert\vec{b}\vert\sin{\theta}\text{ } \)
\( \theta \), is the angle between the tails.
The direction of the cross product vector, \( \hat{n} \), is the normal vector perpendicular to both \( \vec{a}\) and \(\vec{b}\). You can determine the direction of the normal using the right hand rule: Point your right index finger in the direction of the first vector, point your right middle finger in the direction of the second vector. The direction of your thumb is the direction of the normal vector.
1. | \( \vec{a}\times\vec{b} = -\vec{b}\times\vec{a} \) | not commutative |
2. | \( \vec{a}\times(\vec{b}+c)=\vec{a}\times\vec{b}+\vec{a}\times\vec{c} \) | distributive |
3. | \( k(\vec{a}\times\vec{b}) = \vec{a}\times k\vec{b} = k\vec{a}\times\vec{b} \) | multiplication by a scalar |
4. | \( (\vec{a}\times\vec{b})\times\vec{c} \ne \vec{a}\times(\vec{b}\times\vec{c}) \) | not associative |
5. | \( \vec{u}\cdot(v\times\vec{w}) = (\vec{u}\times\vec{v})\cdot\vec{w} \) | triple scalar product - used to prove coplanar if equals zero |
6. | \( \vec{a}\times\vec{b} \) = 0 | collinear vectors (parallel vectors) |
Find \( \vert\vec{u}\times\vec{v}\vert \) for each of the following pairs of vectors. State whether \( \vec{u}\times\vec{v} \) is directed into or out of the page.
Step 1
Step 2
State whether the following expressions are vectors, scalars or meaningless
Remember, cross product produces a vector answer and dot product produces a scalar answer. Both cross and dot product are operations between vectors.