- #1
yashboi123
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- Homework Statement
- What is the cross product of A X C?
- Relevant Equations
- A X B = ABSin(x)
Why are they using Cosine instead of Sine for Cross Product?
- Thread starter yashboi123
- Start date
- #2
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The answer is -96 times sine of angle AOC.
The minus sign arises from the right-hand rule. Take your right hand, point the thumb in the direction of the positive z axis, and curl your fingers. The direction your fingers curl, which is anti-clockwise as we look at the diagram, must be the direction from the vector that's the first argument to the cross product to the vector that's the second argument, in order to get a positive sign. Since in this diagram ##\vec A## is clockwise from ##\vec C## we get a negative sign.
Now what about that cos? The diagram does not mark angle AOC, but we know it it is 90 degrees minus the marked 25 degree angle.
So they just use the formula for sine of the sum of two angles, as follows:
\begin{align*}
\sin\ AOC &= \sin(90 - 25) \\&= \sin(90 + (-25))\\& = \sin 90\ \cos (-25) + \cos 90\ \sin(-25)
\\&= \cos(-25)\times 1 + 0\times \sin(-25)
\\&= \cos\ 25+0
\\&= \cos\ 25\end{align*}
The minus sign arises from the right-hand rule. Take your right hand, point the thumb in the direction of the positive z axis, and curl your fingers. The direction your fingers curl, which is anti-clockwise as we look at the diagram, must be the direction from the vector that's the first argument to the cross product to the vector that's the second argument, in order to get a positive sign. Since in this diagram ##\vec A## is clockwise from ##\vec C## we get a negative sign.
Now what about that cos? The diagram does not mark angle AOC, but we know it it is 90 degrees minus the marked 25 degree angle.
So they just use the formula for sine of the sum of two angles, as follows:
\begin{align*}
\sin\ AOC &= \sin(90 - 25) \\&= \sin(90 + (-25))\\& = \sin 90\ \cos (-25) + \cos 90\ \sin(-25)
\\&= \cos(-25)\times 1 + 0\times \sin(-25)
\\&= \cos\ 25+0
\\&= \cos\ 25\end{align*}
- #3
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@andrewkirk , I would think @yashboi123 is looking for what is wrong with sin(295), following a standard method, rather than for an alternate method. Which method the book used is unknown.
There are some useful formulas:
cos(90-x)=sin(x)=sin(180-x)=-sin(-x)=sin(360+x)
So sin(295)=sin(360-65)=sin(-65)=-sin(65)=-cos(25).
As your calculator will tell you, sin(295)=-cos(25).yashboi123 said:
There are some useful formulas:
cos(90-x)=sin(x)=sin(180-x)=-sin(-x)=sin(360+x)
So sin(295)=sin(360-65)=sin(-65)=-sin(65)=-cos(25).
Related to Why are they using Cosine instead of Sine for Cross Product?
Why is the cosine function used in the cross product formula instead of sine?
Actually, the cross product formula uses the sine function, not the cosine. The magnitude of the cross product of two vectors A and B is given by |A||B|sin(θ), where θ is the angle between the vectors. This is because the cross product measures the area of the parallelogram formed by the vectors, which depends on the sine of the angle between them.
What is the role of sine in the cross product?
The sine function in the cross product formula accounts for the perpendicular component of one vector relative to the other. This perpendicular component is what gives the cross product its direction and magnitude, representing the area of the parallelogram formed by the two vectors.
Why do people mistakenly think cosine is used in the cross product?
Confusion might arise because both sine and cosine functions are fundamental in vector mathematics and trigonometry. While cosine is often used in dot products to measure the projection of one vector onto another, sine is used in the cross product to measure the perpendicularity and the area spanned by the vectors.
How does the sine function affect the direction of the cross product?
The sine function affects the magnitude of the cross product, but the direction is determined by the right-hand rule. The direction of the cross product vector is perpendicular to the plane formed by the original vectors, following the right-hand rule, regardless of the sine function.
Can the cross product be defined using the cosine function?
No, the cross product inherently relies on the sine function to measure the perpendicular component of one vector relative to another. Using the cosine function would not provide the correct geometric interpretation of the area spanned by the vectors, which is essential to the cross product's definition and properties.