Finding the angle between two vectors

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The discussion focuses on the methods for finding the angle between two vectors using sine and cosine rules. It highlights that both sine and cosine can be used interchangeably depending on the context of the problem, though the cosine method is often preferred for its simplicity in calculating the inner product. The example provided illustrates the calculations for angles in both two-dimensional and three-dimensional spaces. Ultimately, the choice between sine and cosine rules may depend on personal preference and the specific requirements of the problem at hand. The consensus is that either method is valid, making the choice flexible.
chwala
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Homework Statement
See attached;
Relevant Equations
sine and cosine angle rules
This is clear to me; i just wanted to know in which contexts is one allowed to use one rule over the other; or it does not matter.

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The angle i realise can also be found by;

##\sin θ = \dfrac{||v×w||}{||v||||w||}##= ##\dfrac{||-3i-5j-11k||}{\sqrt{6}\sqrt{26}}##=##\dfrac{\sqrt{155}}{\sqrt{6}\sqrt{26}}=0.99679## to 5 decimal places...

##⇒θ=\sin^{-1} [0.99679]= 85.41^0##

In which contexts is one allowed to use sine angle rule? ; or is it dependant on the question as directed? cheers...
 
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I picked my own example as follows let;

##p=2i+3j## and ##q=3i+4j## then;

##\cos θ= \dfrac{18}{\sqrt {13}\sqrt{25}}##

##θ = cos^{-1} [0.99846]=3.18^0## to two decimal places

and extending it to ##\mathbb{R^3}## we shall have;

##p=2i+3j+0k## and ##q=3i+4j+0k##

on using cross product we shall end up with,

##\sin θ =\dfrac{1}{\sqrt{13}\sqrt{25}}=0.05547## to 5 decimal places...

##⇒θ=\sin^{-1} [0.05547]= 3.18^0##
 
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chwala said:
Homework Statement:: See attached;
Relevant Equations:: sine and cosine angle rules

or it does not matter.
It does not matter. In most cases I prefer cos because I can calculate inner product easier than vector product.
 
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