1 Physics 111/121 Mini-Review Notes on Vectors

2 Right hand rule: - curl fingers from x to y - thumb points along +z

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4 Multiplying Vectors (Review) Dot Product (Scalar Product)  two vectors  a scalar  measures the component of one vector along the other Cross Product (Vector Product)  two vectors  a third vector normal to the plane they define  measures the component of one vector normal to the other   = smaller angle between the vectors   dot products of Cartesian unit vectors:   cross product of any parallel vectors = zero  cross product is a maximum for perpendicular vectors  cross products of Cartesian unit vectors: i kj Both products distinct from multiplying a vector by a scalar

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10 More About the Cross Product  The quantity AB sin(  is equal to the area of the parallelogram formed by A and B  The direction of C is perpendicular to the plane formed by A and B  The right-hand rule shows the direction.  The distributive rule:  The derivative of a cross product uses the chain rule, but preserving the order of the terms:  Cross products can be calculated using A & B written in terms of the unit vectors (just multiply the terms out, or use determinants).

11 Rotational quantities as vectors: RH Rule Curl fingers of the right hand in the “sense” of the rotational motion Thumb shows direction of a rotational vector quantity, perpendicular to the rotation plane Cross product represents this computationally Picture applies to: displacement  angular velocity  angular acceleration a cross product aXb torque  =rxF angular momentum l=rxp …others…. Right Hand Rule applied to cross product Example: rotation in x-y plane triad of unit vectors x y z 