Finding the angle between two vectors

In summary, the conversation discusses the use of sine and cosine angle rules in different contexts, and whether one is allowed to use one rule over the other or if it does not matter. The conversation also provides examples of using the rules in two and three dimensions. It is ultimately stated that it does not matter which rule is used, with a personal preference towards cosine rule for easier calculation.
  • #1
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.

1675771963098.png


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...
 
Last edited:
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  • #2
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##
 
Last edited:
  • #3
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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Likes MatinSAR, Mark44 and chwala

What is the definition of the angle between two vectors?

The angle between two vectors is the measure of the smallest angle formed between the two vectors when they are placed tail-to-tail.

How do you calculate the angle between two vectors?

The angle between two vectors can be calculated using the dot product formula: θ = cos^-1((a•b)/(|a||b|)), where a and b are the two vectors.

What is the range of values for the angle between two vectors?

The angle between two vectors can have values between 0 and 180 degrees. If the angle is 0 degrees, the vectors are parallel, and if the angle is 180 degrees, the vectors are anti-parallel.

Can the angle between two vectors be negative?

No, the angle between two vectors cannot be negative. It is always measured as a positive value between 0 and 180 degrees.

How does finding the angle between two vectors relate to real-world applications?

Finding the angle between two vectors is a crucial concept in physics and engineering, where it is used to calculate forces and determine the direction of motion. It is also used in navigation, computer graphics, and other fields where vector quantities are involved.

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