Physics Tools Physics for the Life Sciences Lecture #1C Racquetball Striking a Wall Mt. Etna Stewart Hall
More Vector Operations •We know how to add and subtract vectors, either geometrically or by components •What does it mean to multiply two vectors?
The Scalar Product •When we do the “scalar product” we multiply two vectorsto obtain a quantity which is a pure scalar •This is also called the “dot product” or “inner product” •We combine two things with magnitude and direction to get just a magnitude…r F θr F The scalar product r• Fis defined r• F= |r| |F| cos(θ) (magnitude of rtimes magnitude of Ftimes the cosine of the angle in between)Dot product of randF
What does this do for us?
r F θr|| = r cos(θ)r F θF|| = F cos(θ)Note that the dot product is commutative: r • F = F • r = r F cos(θ) Scalar Product Properties r• F = Frcos(θ) = F [rcos(θ)] = ±Fr||r• F = Frcos(θ) = r [Fcos(θ)] = ±r F||
(1) Frcos(θ) (2) ±Fr|| (3) ±rF|| The Big Picture !r⋅!Fyxθ !r!FGiven r, F, and θ, calculate .!r⋅!FFind r!and F!