How do you find the length and direction of vector #-4 - 3i#?

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Answer 1 · 2015-11-21T03:31:56

Length #= 5#

Direction #= (tan^{-1}(frac{3}{4})-pi)# rad counterclockwise from the Real axis.

Let #z=-4-3i#. #z# represents a vector on an Argand diagram.

The magnitude of the vector is the modulus of #z#, which is found using the Pythagoras theorem.

#|z|=sqrt((-4)^2+(-3)^2)=5#

The direction of the vector the principal argument of #z#, which is found using trigonometry.

The basic angle, #alpha=tan^{-1}(frac{3}{4})#.

Since #"Re"(z)<0# and #"Im"(z)<0#, the angle lies in the third quadrant.

#"arg"(z)=-(pi-alpha)#

#=tan^{-1}(frac{3}{4})-pi#