Are the vectors #u=<1, -2># and #v=<4, 8># parallel, orthogonal, or neither?

1 Answer
May 2, 2017

Answer:

Neither parallel nor orthogonal. The angle between the two vectors is:

#theta ~~ 127^@# or #2.21#radians

Explanation:

Given: #vecu=<1, -2># and #vecv=<4, 8>#

Compute the dot-product:

#vecu*vecv= (1)(4)+(-2)(8) = -12#

We computed the dot-product by multiplying the respective components of the vectors. Another definition of the dot-product is the magnitudes of the vectors multiplied by the angle between them:

#vecu*vecv= |vecu||vecv|cos(theta)" [1]"#

Compute the magnitudes of the two vectors:

#|vecu| = sqrt(1^2+(-2)^2) = sqrt(5)#
#|vecv|=sqrt(4^2+8^2) = sqrt(80)#

Substitute the values for the dot-product and the magnitudes into equation [1]:

#-12= sqrt(5)sqrt(80)cos(theta)#

#-12= sqrt(400)cos(theta)#

#-12= 20cos(theta)#

#cos(theta) = -12/20#

#theta = cos^-1(-3/5)#

#theta ~~ 127^@# or #2.21#radians