To determine the magnitude of the difference between two vectors, a and b, we first need to find the vector difference, which is given by:
a – b
Once we have this resulting vector, the magnitude can be calculated using the formula:
|a – b| = √((a1 – b1)² + (a2 – b2)² + … + (an – bn)²)
Here, a1, a2, …, an and b1, b2, …, bn represent the components of vectors a and b, respectively. The magnitude is essentially the length of the resultant vector, which quantifies how far apart the two vectors are in space.
For instance, if vector a has components (3, 4) and vector b has components (1, 2), we would first find the difference:
a – b = (3 – 1, 4 – 2) = (2, 2)
Then, we calculate the magnitude:
|a – b| = √((2)² + (2)²) = √(4 + 4) = √8 = 2√2
Therefore, the value of the magnitude of the difference of vectors a and b precisely quantifies their separation in the vector space.