Assigned work: pg.407 #1-13 Recall dot product produces a scalar from two vectors. Today we examine a Cross Product (or Vector Product) which produces a vector from two vectors.

7.6 Cross Product Cross Product of will be a vector that is perpendicular to both. Therefore the cross product is ONLY defined in R 3. It is useful in physical problems such as torque and area of a parallelogram (applications we will discuss tomorrow)

7.6 Cross Product Prove Cross Product formula :

7.6 Cross Product Cross Product of is the vector:

7.6 Cross Product An easier method to remember the Cross Product formula is:

7.6 Cross Product Finding a Vector Perpendicular to Two Vectors: If are two non-collinear vectors in 3D, then every vector perpendicular to both is of the form where

7.6 Cross Product Ex 1: a)Find a vector perpendicular to the vectors (2,5,0) and (-4,0,9). Answer: (45,-18,20) b)Check your answer using Dot Product (since dot product of 2 perpendicular vectors should be 0).

7.6 Cross Product Magnitude of the Cross Product of is: where is the angle between and

7.6 Cross Product form what we call a right handed system…… Place your right hand on the diagram so that your finger curled in the direction from is an angle less than. The direction of will be the direction your thumb points. Direction of Cross Product of is such that:

7.6 Cross Product Direction of Cross Product of : Thumb is in so:Thumb is out so: is into pageis out of page Fingers curl this way

7.6 Cross Product Direction of Cross Product of : Note there are other methods of this right hand rule to find the direction. Some of you may be used to using the first vector as the thumb, the second vector as your fingers and the direction of the cross product as the palm of your hand. (Motor right hand rule in Grade 11 Physics) Either method works – use which one you like the best.

7.6 Cross Product Ex 2: If and the angle between is 45 degrees Determine the magnitude and direction of. (include a diagram). Answer: Magnitude is and direction will depend on how you drew your diagram.

7.6 Cross Product Ex 3: Find the cross product of : Answer:

7.6 Cross Product Properties of Cross Product: Let be vectors in 3D 1) Order matters (anti-commutative) 2) Distributive Law 3)

7.6 Cross Product Ex 4: Given: Determine: (Note: Think about what order it should be done. Cross product MUST go 1 st since you cannot do cross product of a vector and a scalar) Answer: 78

7.6 Cross Product Important READ 7.6 before doing the assigned questions. Make a note on: 1)What is a “Triple Scalar Product”? 2)What is a Triple Vector Product”?