Compute the dot-product:
< 7, -2, -4 > * < 5, -2, 1 > = (7)(5) + (-2)(-2) + (-4)(1)<7,−2,−4>⋅<5,−2,1>=(7)(5)+(−2)(−2)+(−4)(1)
< 7, -2, -4 > * < 5, -2, 1 > = 35<7,−2,−4>⋅<5,−2,1>=35
Compute the magnitude of both vectors:
|< 7, -2, -4 >| = sqrt(7^2 + (-2)^2 + (-4)^2)|<7,−2,−4>|=√72+(−2)2+(−4)2
|< 7, -2, -4 >| = sqrt(69)|<7,−2,−4>|=√69
|< 5, -2, 1 >| = sqrt(5^2 + (-2)^2 + 1^2)|<5,−2,1>|=√52+(−2)2+12
|< 5, -2, 1 >| = sqrt(30)|<5,−2,1>|=√30
The other way to compute the dot-product is:
< 7, -2, -4 > * < 5, -2, 1 > = |< 7, -2, -4 >||< 5, -2, 1 >|cos(theta)<7,−2,−4>⋅<5,−2,1>=|<7,−2,−4>||<5,−2,1>|cos(θ)
where thetaθ is the angle between the two vectors.
Substitute the values that we have computed:
35 = sqrt(69)sqrt(30)cos(theta)35=√69√30cos(θ)
Solve for thetaθ:
theta = cos^-1(35/sqrt({69}{30}))θ=cos−1(35√{69}{30})
theta ~~ 0.69" radians" θ≈0.69 radians
