Lecture 2.4: Products of Vectors

Learning Objectives

Vectors can be multiplied, not divided.
Two main types of vector products:
- Scalar (Dot) Product: Results in a scalar, used to define work and energy.
- Vector (Cross) Product: Results in a vector, used in defining rotational dynamics like torque.

Definition: Scalar product ( AB = AB \cos \theta ).
- ( \theta ) is the angle between the vectors.
- Dot product is positive for ( 0 < \theta < 90 ), negative for ( 90 < \theta < 180 ).
- Vanishes for orthogonal vectors ( \theta = 90 ).
Properties:
- Commutative: ( AB = BA ).
- Distributive: ( A(B + C) = AB + AC ).
Calculation in Cartesian Coordinates:
- For vectors in component form ( A = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} ), the dot product is: [ AB = A_x B_x + A_y B_y + A_z B_z ]
Example Problems:
- Calculating dot products.
- Finding angles between vectors.

Definition: Vector product ( |AB| = AB \sin \theta ).
- Direction is perpendicular to the plane containing vectors ( A ) and ( B ).
Properties:
- Anticommutative: ( AB = -BA ).
- Distributive: ( A(B + C) = AB + AC ).
Right-Hand Rule: Determines the direction of the cross product.
Application:
- Torque ( \tau = RF ), where ( R ) is the radial vector and ( F ) is the force vector.
Calculation in Cartesian Coordinates:
- For vectors in component form ( A = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} ), the cross product is: [ C = AB = (A_y B_z - A_z B_y) \hat{i} + (A_z B_x - A_x B_z) \hat{j} + (A_x B_y - A_y B_x) \hat{k} ]
Example Problems:
- Calculating cross products.
- Determining torques.
- Magnetic force vector perpendicularity.

Dot product and cross product are distinct mathematical objects with different applications in physics.
Dot product results in a scalar; cross product results in a vector.
Both have specific uses, such as defining work (dot product) and torque (cross product).