How to Prove the Sine Rule Using Cross Product?

In summary, The cross product can be used to prove the sine rule for the area of a triangle with sides a, b, and c. This is done by showing that the cross product of any two sides of the triangle is equal to the cross product of the third side and the sum of the other two sides. This results in the equation absinC = bcsinA = casinB. Dividing by abc, we get the sine formula sinA/a = sinB/b = sinC/c. This can also be demonstrated by taking the cross product of each side and dividing by the magnitude of each vector, resulting in the same formula.
  • #1
iasc
17
0
Could anyone tell me how to use the cross product to prove the sine rule
 
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  • #2
Area of a triangle of side a.b and c is
A = 1/2*axb = 1/2absinC
Similarly 1/2*bxc = 1/2 bcsinA and so on
So
absinC = bcsinA = casinB. Dividing abc to all we get
sinA/a = sinB/b = sinC/c
 
  • #3
Here AB,BC,CA ,a,b,c are vectors and AB=a BC=b CA=c
in a triangle ABC,
AB + BC + CA = 0
a + b + c= 0
a x b =b x c = c x a(proved using above statement just take b and c to other side and take crossproduct with b on both sides first and then with c)
la x bl= |b x c|= |c x a |
|a||b| SinC= |b||c|SinA=|a||c| SinB
dividing by |a||b||c|
we get Sine formula
 
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FAQ: How to Prove the Sine Rule Using Cross Product?

1. What is the definition of the Sine rule using cross product?

The Sine rule using cross product, also known as the Law of Sines, states that the ratio of the sine of an angle in a triangle to the length of the side opposite that angle is constant for all angles and sides in any triangle.

2. How is the Sine rule using cross product applied in real world situations?

The Sine rule using cross product is commonly used in navigation and surveying, as well as in engineering and physics to solve for unknown angles and sides in a triangle.

3. Can the Sine rule using cross product be used for any triangle?

Yes, the Sine rule using cross product can be used for any triangle, regardless of whether it is acute, obtuse, or right-angled.

4. What is the formula for the Sine rule using cross product?

The formula for the Sine rule using cross product is a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the opposite angles.

5. How is the Sine rule using cross product related to the Cross Product Formula?

The Sine rule using cross product is derived from the Cross Product Formula, which states that the cross product of two vectors is equal to the product of their magnitudes and the sine of the angle between them. This relationship allows for the use of the cross product to solve for unknown angles and sides in a triangle.

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