Triangle ABC is an equalateral triangle, with O it's centroid.

a) Show that vector OA + vector OB + vector OC = vector 0

|OA|=|OB|=OC| symmetry

Add components in direction of OA
|OA| - |OB| sin 30 - |OC| sin 30
but sin 30=1/2

Add components perpendicular to OA
|OC|cos 30 - |OB| cos 30 = 0

To establish that vector OA + vector OB + vector OC = vector 0 in an equilateral triangle ABC with centroid O, we can use the properties of centroids.

The centroid O of the equilateral triangle ABC divides each median in a ratio of 2:1. Let M, N, and P be the midpoints of sides BC, AC, and AB, respectively.

1. By definition, the midpoint of a line segment is the average of the coordinates of its endpoints. Therefore, we can express the position vectors of the midpoints as:
vector OM = (vector OB + vector OC) / 2
vector ON = (vector OA + vector OC) / 2
vector OP = (vector OA + vector OB) / 2

2. Since the centroid O divides each median in a 2:1 ratio, we have:
vector OA = 2 * vector OM
vector OB = 2 * vector OP
vector OC = 2 * vector ON

3. Substituting these expressions into the equation vector OA + vector OB + vector OC = vector 0, we get:
2 * vector OM + 2 * vector OP + 2 * vector ON = vector 0

4. Factoring out the common factor of 2, we have:
2 * (vector OM + vector OP + vector ON) = vector 0

5. Finally, dividing both sides by 2, we obtain the desired result:
vector OM + vector OP + vector ON = vector 0

Hence, we have shown that vector OA + vector OB + vector OC = vector 0 for an equilateral triangle ABC with centroid O.

To show that vector OA + vector OB + vector OC = vector 0, we need to use the properties and characteristics of an equilateral triangle.

First, let's define some points. Let A, B, and C be the vertices of the equilateral triangle ABC, and let O be its centroid.

The centroid of a triangle is the point of intersection of its medians. In an equilateral triangle, the medians are concurrent and are also the same as the perpendicular bisectors of the sides. Therefore, the centroid O is the point where the medians intersect, each dividing the triangle into two equal areas.

Now, let's consider vector OA. We can express vector OA as the difference between the position vectors of points A and O. Similarly, vector OB and vector OC can be expressed as the difference between the position vectors of points B and O, and C and O, respectively.

Let's denote the position vectors of points A, B, C, and O as vector a, vector b, vector c, and vector o, respectively.

So, vector OA = vector a - vector o,
vector OB = vector b - vector o, and
vector OC = vector c - vector o.

Substituting these expressions into the equation, we have:
vector OA + vector OB + vector OC = (vector a - vector o) + (vector b - vector o) + (vector c - vector o).

Now, let's group the terms:
= (vector a + vector b + vector c) - 3 * vector o.

In an equilateral triangle, the three sides are equal in length, and the medians divide each other in a 2:1 ratio. Therefore, the sum of the position vectors a, b, and c is equal to zero. This means that vector a + vector b + vector c = vector 0.

Substituting this into the equation, we get:
= vector 0 - 3 * vector o,
= -3 * vector o.

Since -3 * vector o is a scalar multiple of vector o, the sum of vector OA, vector OB, and vector OC is equal to vector 0.

Therefore, vector OA + vector OB + vector OC = vector 0.

In summary, by using the properties and characteristics of an equilateral triangle, we have shown that the sum of the position vectors OA, OB, and OC is equal to vector 0.