How to Add or Subtract Vectors

Vectors are mathematical entities that have both magnitude (or length) and direction. Adding or subtracting vectors involves combining the magnitudes and directions of the vectors in question. Here are the steps to add or subtract vectors:

Adding Vectors: 1. Make sure the vectors being added are of the same dimension (i.e., they have the same number of components). 2. Add the corresponding components of each vector. For example, if you have two vectors A and B, A = [a1, a2, a3] and B = [b1, b2, b3], then the sum of the vectors would be C = [a1 + b1, a2 + b2, a3 + b3]. 3. The resultant vector C is the vector sum of A and B. Its magnitude can be calculated using the Pythagorean theorem: |C| = sqrt(C1^2 + C2^2 + C3^2), where C1, C2, and C3 are the components of C.

Subtracting Vectors: 1. Make sure the vectors being subtracted are of the same dimension. 2. To subtract vector B from vector A, negate the components of B and add them to the corresponding components of A. For example, if A = [a1, a2, a3] and B = [b1, b2, b3], then the difference vector A - B would be C = [a1 - b1, a2 - b2, a3 - b3]. 3. The magnitude of the difference vector C can be calculated using the same formula as for the vector sum: |C| = sqrt(C1^2 + C2^2 + C3^2).

Note that in three dimensions, vectors can be visualized as arrows in space, where the tail of the arrow is at the origin and the head of the arrow is at the point (x,y,z) represented by the vector's components. Adding or subtracting vectors in this way corresponds to placing the vectors end-to-end or tail-to-tail, respectively, and then drawing the resultant vector from the origin to the final point of the arrangement.