Question

Please explain vector addition, triangle method and parallelogram method

Answer 1

Start with 2 arbitrary vectors,
`\vec v_1` and
`\vec v_2`. (pic 1)

Vectors are determined by their lengths and direction. This means that translating the vector (i.e. sliding it left/right and up/down in the plane) doesn't fundamentally change that vector. To this end, we could just as easily represent
`\vec v_2` as if it had originated from the tip of
`\vec v_1`. This "new"
`\vec v_2` and the "old"
`\vec v_2` are the same vector. (pic 2)

If we connect the origin of
`\vec v_1` with the tip of "new"
`\vec v_2`, we get a new vector, and this we define as the vector sum
`\vec v_1+\vec v_2`. (pic 3)

We can do this other way, by first traslating
`\vec v_1` to the tip of
`\vec v_2`, then connecting the origin of
`\vec v_2` with the tip of "new"
`\vec v_1`. This demonstrates that vector addition is commutative (order of the vectors being added doesn't matter - you always end up at the same terminus). The "parallelogram method" refers to how a parallelogram is traced out. (pic 4)

Multiplying a vector by -1 reverses its direction. (pic 5)

Adding
`\vec v_1` and
`-\vec v_2` works the same way as standard vector addition, giving us the new vector
`\vec v_1-\vec v_2`. (pic 6)

We can do the same in the reverse order, but now we get a different vector,
`\vec v_2-\vec v_1`. (pic 7)

These vectors have the same length but point in opposite directions. (pic 8)

But notice that we can translate the vectors
`\vec v_1-\vec v_2` and
`\vec v_2-\vec v_1` so that we get a vector that either starts at the tip of
`\vec v_2` and ends at the tip of
`\vec v_1` (pic 9), or starts at the tip of
`\vec v_1` and ends at the tip of
`\vec v_2` (pic 10). The "triangle method" refers to the triangles that are traced out by either vector sum
`\vec v_1-\vec v_2` and
`\vec v_2-\vec v_1` together with
`\vec v_1` and
`\vec v_2`.

Answer 2

vector addition is adding two or more vectors together
triangle method is when your getting to two numbers from the triangle and adding it to get the missing side of the triangle
parallelogram method is when your drawing a triangle connecting the vectors from head to tail