1. IB HL www.ibmaths.com Adrian Sparrow Vectors: the cross product

2. Introduction to the cross product The cross product of two vectors is written as v x w. The cross product, v x w, will produce a new vector that is perpendicular to both vectors v and w. This will work for vectors that intersect. Or for vectors that are skew.

3. Calculating the cross product From the formula sheet the cross product is defined as: For example: Do not confuse this with the dot product.

4. Questions

5. The area of a triangle From the formula sheet: The proof of this:Expand, simplify and factorize: This is useful as the area of a triangle is:

6. Finding area of a triangle Calculate the area of the triangle with vertices A(1,2,3), B(2,0,-3) and C(2,1,-1). Let vector v=AB and w=AC.

7. Shortest distance between two skew lines Find the shortest distance between the following 2 skew lines in space. Rearrange the vectors. The cross product of the two vectors is perpendicular to both vectors. Take the vector part of each line and find the cross product. The points on L 1 and L 2 where the lines are closest will have a vector between the points that is parallel to the cross product.

8. Shortest distance between two skew lines PQ is parallel to the cross product. Set up simultaneous equations and solve. 1 2

9. Shortest distance between two skew lines Solve the simultaneous equations. And finally.... Ready to do one on your own...

10. Shortest distance between two skew lines Find the shortest distance between the following 2 skew lines in space. 1. Find the vector part of the lines and then the cross product. 2. Find two general points, P and Q, on the skew lines and form the vector between them. 3. Make this parallel to the cross product, form simultaneous equations and find u and t. 4. Now find the modulus of vector PQ.