- #1
yashboi123
- 17
- 0
- 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
- 4,119
- 1,717
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
- 41,394
- 9,870
@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).
1. Why is Cosine used in the Cross Product formula instead of Sine?
The use of Cosine in the Cross Product formula is due to the geometric interpretation of the Cross Product. The Cross Product of two vectors gives a vector that is perpendicular to both of the original vectors. This perpendicular vector can be calculated using the dot product of the two original vectors and the cosine of the angle between them. This is why Cosine is used in the Cross Product formula instead of Sine.
2. Can Sine be used in place of Cosine in the Cross Product formula?
No, Sine cannot be used in place of Cosine in the Cross Product formula. This is because the Cross Product formula is derived from the geometric interpretation of the Cross Product, which involves the use of Cosine. Using Sine in the formula would result in a different vector that is not perpendicular to the original vectors.
3. What is the mathematical explanation for using Cosine in the Cross Product formula?
The mathematical explanation for using Cosine in the Cross Product formula is that the dot product of two vectors can be calculated using the magnitude of the vectors and the cosine of the angle between them. This dot product is then multiplied by the unit vector perpendicular to the plane containing the two original vectors, resulting in the Cross Product vector.
4. Are there any advantages to using Cosine instead of Sine in the Cross Product formula?
Yes, there are advantages to using Cosine instead of Sine in the Cross Product formula. One advantage is that the Cross Product vector calculated using Cosine will always be perpendicular to the original vectors, while using Sine may result in a vector that is not perpendicular. Additionally, the use of Cosine in the formula simplifies the calculation and makes it more efficient.
5. Are there any real-world applications where using Cosine in the Cross Product formula is important?
Yes, there are many real-world applications where using Cosine in the Cross Product formula is important. Some examples include calculating torque in physics, determining the direction of magnetic fields, and calculating the angular momentum of rotating objects. In these applications, the use of Cosine ensures accurate and meaningful results.
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