What is the angle between <-3,9,-7 > and < 4,-2,8 >?

1 Answer
Feb 20, 2016

theta~= 2.49 radians

Explanation:

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