Final answer:
Using the dot product and the magnitudes of the vectors, we calculated the angle between vectors u=⟨3,4⟩ and v=⟨−1,−2⟩. The calculated angle is approximately 12 degrees, which does not match any of the options given in the question. This does not correspond to any of the options A)
Step-by-step explanation:
To find the angle between two vectors u and v, we use the dot product formula and the magnitude of each vector. The dot product of u and v is calculated as follows:
u · v = (3)(-1) + (4)(-2) = -3 - 8 = -11
The magnitude (or length) of vector u is:
||u|| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5
The magnitude of vector v is:
||v|| = √((-1)^2 + (-2)^2) = √(1 + 4) = √5 = √5 ≈ 2.236
Now, to find the angle θ between vectors u and v, we use the formula:
cos(θ) = (u · v) / (||u|| · ||v||)
cos(θ) = -11 / (5 · 2.236) ≈ -0.982
θ = cos^{-1}(-0.982) ≈ 168º
However, we are looking for the smallest angle between the two vectors which is the supplement of 168º. Therefore:
180º - 168º = 12º
The smallest angle between vectors u and v is approximately 12º. This does not correspond to any of the options A) 148º, B) 143º, C) 131º, or D) 127º provided in the question, so please double-check the choices or the initial vectors provided.