The cross product
The cross product (sometimes also called the vector product ) is defined as follows: (a1a2a3)×(b1b2b3)=(a2b3−b2a3−(a1b3−b1a3)a1b2−b1a2) It is a bit unwieldy: we should practice the computation until we can reproduce it accurately.
Normal vectors and areas
There are 2 main reasons for introducing the cross product. The first is that the result of the cross product, n=a×b is a vector that is perpendicular to both a and b. This is sometimes called the normal vector and will be used extensively in chapter 11c when we work with planes in 3 dimensions.
The second is that the cross product leads to a formula very similar to the dot product formula we encountered in the previous section:
where θ is the angle between vectors a and b. Notice the two differences between this and the dot product formula: first we take the magnitude of a×b on the LHS of the equation, and second we have a sine instead of a cosine.
The derivation of the formula is a bit out of our scope so we will take the formula as is. Hopefully the formula doesn't sound too foreign to us: we have seen it in the calculation of areas of triangles and parallelograms in previous study. This gives us two formulas:
Solution to examples
Find a vector n, such that n is perpendicular to →OA and →OB.
Solution: n=(12−3)×(501)=(2−16−10)
Remark: Any parallel vector will work as an answer.
Solution: Area of triangle OAB=12|a×b|=12|2i−16j−10k|= 3√10 units2.