Dot Product of Equilateral Triangle

In summary, the dot product of two unit vectors can be found by taking the cosine of the angle between them.
  • #1
mill
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Homework Statement



In an equilateral triangle with sides u, v, w, where they are all unit vectors, find u dot w.

Homework Equations



u dot w = |u||w|cosθ

The Attempt at a Solution



The answer is ##\frac {-1} {2} ##

cos(120) = -1/2

Elsewhere, I read the statement that since these are already unit vectors then the dot products are simply the angles between them. What does this mean? Does something cancel out the |u||w| in the denominator? Or do |u||w| function to make the dot in terms of unit vectors so they would both be 1 in this case?

Nevermind, got it.
 
Last edited:
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  • #2
mill said:
Elsewhere, I read the statement that since these are already unit vectors then the dot products are simply the angles between them.

...the dot products are the cosine of the angle between them.
 

1. What is the formula for calculating the dot product of an equilateral triangle?

The formula for calculating the dot product of an equilateral triangle is:
dot product = a^2 + b^2 + c^2, where a, b, and c are the sides of the triangle.

2. How is the dot product of an equilateral triangle used in geometry?

The dot product of an equilateral triangle is used to calculate the magnitude of a vector that is perpendicular to the triangle's plane. It is also used to find the angle between two vectors.

3. Can the dot product of an equilateral triangle be negative?

No, the dot product of an equilateral triangle cannot be negative. This is because all three sides of an equilateral triangle have the same length, so the formula for the dot product will always result in a positive value.

4. Is the dot product of an equilateral triangle the same as the dot product of any other triangle?

No, the dot product of an equilateral triangle is not the same as the dot product of any other triangle. This is because the sides of an equilateral triangle are all equal, while the sides of other triangles can have different lengths.

5. How is the dot product of an equilateral triangle related to the Law of Cosines?

The Law of Cosines states that the square of a triangle's side is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle. This can also be written as a^2 = b^2 + c^2 - 2bc*cos(A), where A is the angle between sides b and c. The dot product of an equilateral triangle can be used to calculate this cosine value, making it related to the Law of Cosines.

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