What is the angle between #<7 , 2 , -3 > # and # < 8 , 6 , 0 > #?

1 Answer
Apr 19, 2016

approximately #0.528 " radians"# or #30.28^@#

Explanation:

The angle, #theta#, between two vectors #u# and #v#
is such that
#color(white)("XXX")cos(theta)=(u*v)/(abs(abs(u))*abs(abs(v)))#

For
#color(white)("XXX")u=< u_1,u_2,u_3 > = < 7,2,-3 >#
and
#color(white)("XXX")v=< v_1,v_2,v_3 > = < 8,6,0 >#

#u*v= u_1*v_1+u_2*v_2+u_3*v_3#
#color(white)("XXx")=7xx8+2xx6+(-3)xx0=68#

#abs(abs(u)) = sqrt(u_1^2+u_2^2+u_3^2)#
#color(white)("XXx")=sqrt(7^2+2^2+(-3)^2)=sqrt(62)#

#abs(abs(v))=sqrt(v_1^2+v_2^2+v_3^2)#
#color(white)("XXx")=sqrt(8^2+6^2+0^2)=sqrt(100)=10#

Therefore
#color(white)("XXX")cos(theta)= 68/(10sqrt(62))#
#color(white)("XXXXXXX")~~0.863601# (using a calculator)
and
#color(white)("XXX")theta=arccos(0.863601) ~~ 0.528426# (radians)