Final answer:
To find the angle between two vectors (-1,2) and (2,-4), we calculate their dot product and use the inverse cosine formula. However, in this case, the calculated value is outside the valid range, suggesting an error in the computations or understanding of the question.
Step-by-step explanation:
To find the angle between two vectors given by their Cartesian coordinates, we can use the dot product formula. In this case, we are looking for the angle between the vectors (-1,2) and (2,-4). The dot product of these vectors is (-1)×(2) + (2)×(-4) = -2 - 8 = -10. The magnitudes (|A| and |B|) of the vectors are |A| = √((-1)² + (2)²) = √5 and |B| = √((2)² + (-4)²) = √20.
The formula to find the angle θ between two vectors is given by the cosine of the angle which is equal to the dot product of the vectors divided by the product of their magnitudes:
θ = cos⁻¹((∑A_iB_i) / (|A|×|B|))
Plugging our values into this formula, we get:
θ = cos⁻¹((-10) / (√5 × √20))
θ = cos⁻¹((-10) / (5))
θ = cos⁻¹(-2)
Since the value -2 is outside the range of the cosine function, there must be an error in calculation or a misinterpretation of the vectors. The actual angle calculation would involve an arccosine (inverse cosine) of a value between -1 and 1, which reflects the possible range of the dot product of two normalized vectors.