If u is a unit vector, how do you find the vectors u, v, and w, assuming that v and w are also unit vectors forming an equilateral triangle?

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To solve this problem, we need to first understand the properties of unit vectors and the geometric arrangement they form. A unit vector is defined as a vector with a magnitude of 1. When we have three unit vectors forming an equilateral triangle, there are specific relationships between these vectors that we can use.

Step 1: Define the Unit Vector u

Let’s assume our unit vector u is defined in a two-dimensional space. For example, we can represent u as:

u = (1, 0)

Step 2: Determine the Positions of v and w

Since we are forming an equilateral triangle with the unit vector u, we need to determine the vectors v and w. In an equilateral triangle, the angles between each pair of vectors are all 60 degrees.

To find v and w, we can use rotation matrices. Rotating vector u around the origin by 60 degrees to find v gives:

v = (cos(60°), sin(60°)) = (0.5, √3/2)

Next, to find w, we can rotate vector u by -60 degrees:

w = (cos(-60°), sin(-60°)) = (0.5, -√3/2)

Step 3: Verifying the Unit Vectors

Now we should check that v and w are indeed unit vectors:

||v|| = √(0.5² + (√3/2)²) = √(0.25 + 0.75) = √1 = 1
||w|| = √(0.5² + (-√3/2)²) = √(0.25 + 0.75) = √1 = 1

Both vectors v and w are confirmed to be unit vectors.

Final Result

Thus, the unit vectors u, v, and w that form an equilateral triangle can be represented as:

u = (1, 0)
v = (0.5, √3/2)
w = (0.5, -√3/2)

These vectors will create an equilateral triangle in two-dimensional space, maintaining the properties of unit vectors.

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