Note: The angel between two nonzero vector u and v, where #0 <= theta <= pi# is define as
#vec u = < u_1, u_2,u_3 > #
#vec v = < v_1, v_2,v_3 > #
#cos theta= (u *v )/(||u|| " ||v|| #
Where as: #" " u *v= (u_1v_1) + (u_2v_2) + (u_3v_3)#
#||u|| = sqrt((u_1)^2 +(u_2)^2+(u_3)^2)#
#||v||=sqrt((v_1)^2 +(v_2)^2+(v_3)^2)#
Step 1: Let
#vec u = < -3, 9, -7 > # and
#vec v= < 4, -2, 8 > #
Step 2: Let's find #color(red)(u *v)#
#color(red)(u *v) = (-3)(4) + (9)(-2) + (-7)(8)#
#= -12 -18 -56#
#= color(red)(-86)#
Step 3: Let find #color(blue)(||u||)#
#vec u= < -3, 9 - 7>#
#color(blue)(||u||) = sqrt((-3)^2 + (9)^2 + (-7)^2)#
#=sqrt(9+81+49)#
#=color(blue)(sqrt139)#
Step 4 Let find #color(purple)(||v||)#
#vec v = < 4, -2, 8>#
#color(purple)(||v||) = sqrt((4)^2 + (-2)^2 + (8)^2)#
#= sqrt(16 + 4 + 64)
=color(purple)(sqrt84)#
Step 5; Let substitute it back to the formula given above, and find #theta#
#cos theta= (u *v)/(||u|| " ||v||) #
#cos theta = color(red)(-86)/((color(blue)sqrt(139))color(purple)((sqrt84))#
#cos theta = color(red)(-86)/(sqrt11676)#
#theta= cos^(-1)(-86/(sqrt11676))#
#theta~= 2.49# radians
**note: this is because #u *v <0#
