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

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The discussion explores the application of the Cosine Rule to vector addition in three-dimensional space. It confirms that the formula for the magnitude of the sum of two vectors, ||v+w||^2, can be expressed similarly to the Cosine Rule, with adjustments for the angle between the vectors. By substituting one vector with its negative, the angle shifts by 180 degrees, affecting the cosine term accordingly. A parallelogram diagram is suggested to visualize the relationship and apply the Cosine Rule effectively. The conversation emphasizes the validity of using the Cosine Rule in this context.
WMDhamnekar
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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 $\mathbb{R}^3$
 
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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.
 
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 $\theta$".
 

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