MHB Does the Cosine Rule Apply to Vector Addition in 3-D?

[WMDhamnekar]

Hi,
In $\mathbb{R^3} || v-w ||^2=||v||^2 + ||w||^2 - 2||v||\cdot ||w||\cos{\theta}$ But can we say $||v+w||^2=||v||^2 +||w||^2 + 2||v|| \cdot||w|| \cos{\theta}$ where v and w are any two vectors in $\mathbb{R}^3$

 

[HOI]

Replace w with -w. Since that reverses the direction of w, it adds 180 degrees to θ . cos(θ+ 180)= cos(θ)cos(180)- sin(θ)sin(180)= cos(θ)(-1)+ sin(θ)(0)= -cos(θ). Yes, that just changes the sign on the last term.

 

[WMDhamnekar]

Hi,
One math expert provided the following answer. " Draw a parallelogram diagram. Apply the cosine rule using angle φ which is the complementary angle to $\theta$".