The force on an object is F= −10j. For the vector v =−5i −j , find A) The component of F parallel to v B) The component of F perpendicular to v

Answer 1

It is helpful to note that F = <0, -10> and V = <-5, -1> in the i, j domain. If a variable is bold or in italics then it is a vector. Also, note that for the vector <a, b> that |<a, b>| = (a² + b²)^1/2 (this is the "magnitude" of a vector).

Part A:

The component of F parallel to V is the same thing as the vector projection of F on V, which is given by the formula Proj_VF = (F⋅V / |V|²)V. This is really Proj_VF = F⋅v², where v is the unit vector of V, which follows v = V / |V|. If these formulas makes no sense, I would recommend reading up on vector projection and unit vectors. Using this equation:

Proj_VF = (F⋅V / |V|²)V

Proj_VF = [<0, -10> ⋅ <-5, -1> / ((-5)² + (-1)²)]<-5, -1>

Proj_VF = [10 / 26]<-5, -1>

Proj_VF = <-25/13, -5/13>

Proj_VF = -25i/13 - 5j/13 << final answer to Part A

Part A:

The component of F perpendicular to V is the same thing as F - Proj_VF:

F - Proj_VF = (-10j) - (-25i/13 - 5j/13)

F - Proj_VF = 25i/13 -125j/13 << final answer to Part B

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