In an equilateral triangle with sides of length 1 represented by vectors v and w, how do you compute the dot product v · w?

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To compute the dot product of vectors v and w in an equilateral triangle, we start by understanding the properties of the triangle and the vectors involved.

In an equilateral triangle, all sides are of equal length, which in this case is 1. The angles between each pair of sides are all 60 degrees. Assuming that vector v is directed along one side of the triangle, we can express vector v as:

v = (1, 0)

For vector w, which represents another side of the triangle, we need to account for the 60-degree angle. We can express w in terms of its components, utilizing trigonometry:

w = (cos(60°), sin(60°))

Knowing that:

  • cos(60°) = 1/2
  • sin(60°) = √3/2

We can now express vector w as:

w = (1/2, √3/2)

Next, we compute the dot product v · w using the formula:

v · w = vxwx + vywy

Substituting in our values:

v · w = (1)(1/2) + (0)(√3/2) = 1/2 + 0 = 1/2

Therefore, the dot product of vectors v and w is:

v · w = 1/2

This result indicates a non-orthogonal relationship between the two vectors, consistent with the properties of an equilateral triangle.

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