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.

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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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  • #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##
 
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  • #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

FAQ: Finding the angle between two vectors

What is the formula to find the angle between two vectors?

The formula to find the angle between two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the dot product formula: \[ \cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]where \( \mathbf{A} \cdot \mathbf{B} \) is the dot product of the vectors, and \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of the vectors. The angle \( \theta \) can then be found using the inverse cosine function: \[ \theta = \cos^{-1}\left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}\right) \]

What is the dot product of two vectors?

The dot product of two vectors \( \mathbf{A} = [A_1, A_2, A_3] \) and \( \mathbf{B} = [B_1, B_2, B_3] \) in three-dimensional space is calculated as:\[ \mathbf{A} \cdot \mathbf{B} = A_1 B_1 + A_2 B_2 + A_3 B_3 \]In general, for vectors in \( n \)-dimensional space, the dot product is:\[ \mathbf{A} \cdot \mathbf{B} = \sum_{i=1}^{n} A_i B_i \]

How do you find the magnitude of a vector?

The magnitude of a vector \( \mathbf{A} = [A_1, A_2, A_3] \) in three-dimensional space is found using the Euclidean norm:\[ |\mathbf{A}| = \sqrt{A_1^2 + A_2^2 + A_3^2} \]For a vector in \( n \)-dimensional space, the magnitude is:\[ |\mathbf{A}| = \sqrt{\sum_{i=1}^{n} A_i^2} \]

Can the angle between two vectors be negative?

No, the angle between two vectors cannot be negative. The angle \( \theta \) calculated using the inverse cosine function \( \cos^{-1} \) will always yield a value between 0 and 180 degrees (or between 0 and \( \pi \) radians). This is because the cosine function ranges from -1 to 1, and the inverse cosine function maps these values to angles within the specified range.

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