1. Vectors and Scalars
AP Physics C

2. The “Cross” Product (Vector Multiplication)
Multiplying 2 vectors sometimes gives you a VECTOR quantity which we call the VECTOR CROSS PRODUCT.
In polar notation consider 2 vectors:
A = |A| < θ1 & B = |B| < θ2
The cross product between A and B produces a VECTOR quantity. The magnitude of the vector product is defined as:
Where q is the NET angle between the two vectors. As shown in the figure. q A B

3. The Vector Cross Productq A B
30k, curl moves counter clockwise

4. Cross Products and Unit Vectors
The cross product between B and A produces a VECTOR of which a 3x3 matrix is need to evaluate the magnitude and direction.
You start by making a 3x3 matrix with 3 columns, one for i, j, & k-hat. The components then go under each appropriate column.
Since B is the first vector it comes first in the matrix

5. Cross Products and Unit Vectors
You then make an X in the columns OTHER THAN the unit vectors you are working with.
For “i” , cross j x k
For “j” , cross i x k
For “k” , cross i x j
Let’s start with the i-hat vector: We cross j x k
k=(BxAy)-(ByAx)
Now the j-hat vector: We cross i x k
Now the k-hat vector: We cross i x j

6. Cross Product Restated
A x B = (AyBz – AzBy)i – (AxBz-AzBx)j + (AxBy – AyBx)

7. Example Let’s start with the i-hat vector: We cross j x k
Now the j-hat vector: We cross i x k
-44i-8j+20k
Now the k-hat vector: We cross i x j
The final answer would be:

8. Example 1 Calculate the cross product of A=2ĩ+3ĵ and B=-ĩ+ĵ+4ḱ.
A x B= (3(4)-0(1))i – (2(4) -0(-1))j + (2(1) -3(-1))k
A x B = 12i -8j + 5k
1. 12i-8j+5k

9. Example 2
2. Two vectors lying in the xy plane are given by the equations A=2ĩ+3ĵ and B=-ĩ+2ĵ.
a) Calculate A x B = 7k
b) Calculate B x A = -7k
Find the angle between A and B
|A x B| = 7
|A|=√(22 +32) = √13
|B|=√(12 +22)=√5
|AxB|=|A||B|sinᶱ ᶿ=60.3 degrees
2. A. 7k b. -7k c degrees

10. Example 3
A particle is located at the vector position r = (i + 3j)m, and the force acting on it is F = (3i + 2j) N. What is the torque on the particle about an axis through the origin?
Ʈ=r x F = (3(3)-1(2))k =7 Nm k -7

11. Example 4 Find A x B for the following choices: A = 4i and B=6i +6j
A=4i and B=6i +6k
A=2i+3j and B=3i + 2j
A x B = 24k
A x B = 24j
A x B =5k

12. Torque
Torque is the cross product of the radial displacement vector and force ԏ=r x F

13. Angular Momentum
Angular Momentum of a particle is defined as the cross product of the radial vector and the linear momentum: L=r x p

14. Torque and Angular Momentum