Why use cos for dot product and sin for cross product?

In summary, the dot product and cross product are complementary operations used in vector calculations. The dot product calculates the projection of one vector onto another, while the cross product gives a vector normal to both. The dot product is related to the projection of one vector onto another, while the cross product is related to the area of a parallelogram formed by two vectors.
  • #1
Mamoon
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0
Dose anybody knw that why we take cos with dot product and Sin with cross product?
 
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  • #2
Welcome to PF!

They are complementary operations.Tthe dot product gives you the part of vector A projected onto B whereas the cross product gives you the part of A not projected onto B and vice versa.

A dot B = |A| |B| cos(AB) and the project of A on B = |A| cos (AB) = (A dot B) / |B|

The cross product also gives you a vector normal to both A and B using the righthand rule by convention.

There are other geometric ways of looking at it too. The cross product is the area of the parallelogram with A and B as its sides.
 
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  • #3
The dot product is related to the projection of one vector on another. If you draw vectors u and v with "ends" together and drop a perpendicular from the tip of vector u to vector v, then you have a right triangle in which the length of u is the hypotenuse and the length of the projection is the "near side".
 

1. What is the significance of sin and cos in mathematics?

Sin and cos, also known as sine and cosine, are two important trigonometric functions used in mathematics to describe the relationships between the angles and sides of a right triangle. These functions are used extensively in fields such as physics, engineering, and astronomy to solve problems involving triangles and periodic phenomena.

2. How are sin and cos related to each other?

Sin and cos are related to each other through the Pythagorean identity, which states that sin²θ + cos²θ = 1. This means that for any angle θ, the square of the sine of θ plus the square of the cosine of θ will always equal 1.

3. What are the applications of sin and cos in real life?

Sin and cos have numerous applications in real life, such as in navigation and surveying, where they are used to calculate distances and angles. They are also used in the fields of music and sound engineering to describe the periodic nature of sound waves. Additionally, they are used in computer graphics to create smooth curves and animations.

4. Can sin and cos be negative?

Yes, sin and cos can be negative. The sine function can take on values between -1 and 1, while the cosine function can take on values between -1 and 1. The sign of sin and cos depends on the quadrant in which the angle falls, with positive values in the first and fourth quadrants and negative values in the second and third quadrants.

5. How are sin and cos used to solve trigonometric equations?

Sin and cos can be used to solve trigonometric equations by applying the inverse trigonometric functions, such as arcsin (sin⁻¹) and arccos (cos⁻¹), which allow us to find the angle given the value of the sine or cosine. These inverse functions are essential in solving problems involving triangles and periodic functions in mathematics and other fields.

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