A.B = AB cos(x), and AxB = AB sin(x)

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In summary, the conversation discusses the difference between scalar and vector products, specifically the formulas A.B = AB cos(x) and AxB = AB sin(x). The speaker raises the issue of not knowing which formula to use in a given problem and suggests understanding the problem better can help determine the appropriate product to use.
  • #1
newton1
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i know A.B = AB cos(x), and AxB = AB sin(x) ...
but if we face the problem didn't told us what should we use
then how to know what should we use
how to different scalar product or vector product...??
 
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  • #2
Understanding the problem might help!


It might be an important point that "AxB = AB sin(x)" is not true.
AxB is a vector whose length is the length of A times the length of B times sin(x). If you made a little more effort to distinguish between vectors and their lengths in your writing, it might help to decide whether the product in your problem was to result in a vector or a scalar!:smile:
 
  • #3


When faced with a problem that does not specify which operation to use, it is important to understand the difference between scalar product and vector product and how they are used in different situations.

A scalar product, also known as dot product, is a mathematical operation that results in a scalar quantity. It is used to calculate the projection of one vector onto another and is represented by the symbol "·" or by parentheses "( )". The result of a scalar product is a single number and is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. In the context of geometry, the scalar product is used to calculate the work done by a force or the angle between two vectors.

On the other hand, a vector product, also known as cross product, is a mathematical operation that results in a vector quantity. It is used to calculate the direction and magnitude of a vector perpendicular to two other vectors and is represented by the symbol "x". The result of a vector product is a vector and is calculated by multiplying the magnitudes of the two vectors and the sine of the angle between them. In the context of geometry, the vector product is used to calculate the torque or the area of a parallelogram.

To determine which operation to use, it is important to carefully read and understand the problem and the given information. If the problem involves calculating the projection or work done, then the scalar product should be used. If the problem involves calculating the direction and magnitude of a vector perpendicular to two other vectors or the torque or area of a parallelogram, then the vector product should be used.

In summary, understanding the difference between scalar product and vector product and when to use each one is crucial in solving mathematical problems. It is important to carefully read and analyze the given information to determine which operation to use.
 

1. What is the difference between A.B and AxB in the equation AB cos(x) and AB sin(x)?

Both A.B and AxB are products of two vectors A and B. However, in the equation AB cos(x), A.B represents the dot product of vectors A and B, while in the equation AB sin(x), AxB represents the cross product of vectors A and B.

2. How do you calculate the dot product and cross product of two vectors?

The dot product of two vectors A and B is given by A.B = |A||B| cos(θ), where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them. The cross product of two vectors A and B is given by AxB = |A||B| sin(θ), where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.

3. What is the significance of the angle (θ) in the equations AB cos(x) and AB sin(x)?

The angle (θ) represents the angle between the two vectors A and B. This angle determines the magnitude of the dot product and cross product of the vectors, and thus plays a crucial role in the overall calculation.

4. Can the equations AB cos(x) and AB sin(x) be used for all types of vectors?

Yes, these equations can be used for any type of vector, including 2D and 3D vectors. However, the vectors must be in the same dimension for the dot product or cross product to be calculated.

5. What are some real-world applications of the equations AB cos(x) and AB sin(x)?

The dot product and cross product of vectors have numerous applications in physics, engineering, and mathematics. Examples include calculating torque, work, and power in mechanics, determining electric and magnetic fields in electromagnetism, and analyzing forces and moments in structural analysis.

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