Vectors and Scalars AP Physics C

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

The Vector Cross Product q A B 30k, curl moves counter clockwise

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

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

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

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:

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

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. 60.3 degrees

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

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

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

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

Torque and Angular Momentum