1 Cross Product Basics 1.1 Denition and Basic Calculations Tile cross product lets us multiply vectors together to get another vector. I'll dene it by writing down its multiplication table for the basic unit vectors i,j,k: % 0 jxi: kx1=1 Reading the circle clockwise, the rst two basis vectors you read are the ones being multiplied and the third one is the result. For example, starting with i we get to} and then it. Therefore, 2 X j If you read the circle counterclockwise, then the result of the multiplication is negative: 3 X i = k. Question 1.1) Using the multiplication table or the product circle and the distributive law of multiplication', calculate the following cross products: 1. (2233) x (44-23) Example: (2:733) x (#423) =4(i x3) +3 (3 xi) =12 2. (371%) x (2H3) 'ax(b+c):nxb+axc Question 1.2) Calculate the dot products of your answers from the previous questions with Question 1.5) Calculate A x B for the vectors displayed in the previous problem in terms the vectors being multiplied. of the magnitudes A and B and the angle 0 between them using the technique described 1. ( 21 - 3]) , (-i+2j) earlier in this section. Example: The answer for the cross product was k. Dotting that into the two vectors gives: (21 - 3]) .k= 0 ( -i+2]) .k = 0 2. (3 - k) , (22 + 3 ) Question 1.6) Use your result above to explain why A x B = A|B sin0, is true for any two vectors with angle 0 between them. [Hint: You are free to choose your x, y, z coordinates in any orientation you want. 3. (itith ) , (i - 3 -k) Question 1.3) What do your results from the previous problem imply about the angle between the vectors being multiplied and the vector we get from their cross product? Question 1.4) Consider the vectors A and B lying in the x - y plane as depicted in the image above. The vector A= A i. Write down the components of B using its magnitude and the angle 0. 2 3