I am doing vector diagrams with two vectors and you have to find the direction and magnitude of the resultant vector. How do I know which angle to use to describe the direction of the vector?

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Answer 1 · 2018-06-19T09:18:39

See below.

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From the diagram.

We have vectors #bba# and #bb(b)#, with #bbc# being the resultant vector. ( #bbc=bba+bb(b)#)

Writing the vectors in column form.

#bba=((2-1),(4-1))=((1),(3))#

#bb(b)=((4-2),(6-4))=((2),(2))#

Resultant vector #bbc# is the sum of #bba# and #bb(b)#

#:.#

#bbc=((1),(3))+((2),(2))=((1+2),(3+2))=((3),(5))#

The magnitude of some vector #bbv#, #bbv=((x),(y))# is:

#||bbv||=sqrt(x^2+y^2)#

This is just Pythagoras's theorem. You have probably also seen this as the distance formula.

So magnitude of resultant vector #bbc#:

#||bbc||=sqrt((3)^2+(5)^2)=sqrt(34)#

The direction is given as the angle the vector makes with the positive x axis. This is given as a positive angle .i.e. anti-clockwise rotation.

This angle can be found using the tangent ratio.

Since the tangent ratio is:

#tan(theta)=("opposite")/("adjacent")#

This is the same as:

#tan(theta)=(y/x)#

So for vector #bbc#:

#tan(theta)=5/3#

Therefore:

#theta=arctan(5/3)=59^@# nearest degree.

This is the most common form of direction, but there are situations where the direction can be given as a bearing.