To solve the problem, we start by recognizing what it means for vectors to be unit vectors. A unit vector is a vector that has a magnitude of 1. Let’s denote the vector u as a unit vector. We also have two other unit vectors v and w, which together with u form an equilateral triangle.
In an equilateral triangle, the angle between any two vectors is 60 degrees (or π/3 radians). Thus, we can use the dot product formula to find u · v and u · w.
The dot product of two vectors a and b is given by:
a · b = |a| |b| cos(θ)
Where:
- |a| and |b| are magnitudes (lengths) of vectors a and b.
- θ is the angle between them.
Since all vectors u, v, and w are unit vectors, their magnitudes are all equal to 1:
|u| = |v| = |w| = 1
Now, applying this to the dot products:
u · v = |u| |v| cos(60°) = 1 * 1 * cos(60°) = 1 * 1 * (1/2) = 1/2
Similarly, for u · w:
u · w = |u| |w| cos(60°) = 1 * 1 * cos(60°) = 1 * 1 * (1/2) = 1/2
Therefore, the results are:
- u · v = 1/2
- u · w = 1/2
In conclusion, when u, v, and w are unit vectors that form an equilateral triangle, the dot products u · v and u · w yield 1/2 each.