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Answer :
By using the dot product and the magnitude of vectors, the angle θ between the vectors u and v is calculated to be approximately 37 degrees, making option 3) the correct choice.
Let's find the angle θ between the vectors u = -9i+4j and v = -5i+9j. To do this, we use the dot product of vectors and the formula for the angle between two vectors:
cos(θ) = (u \\cdot v) / (|u| \\cdot |v|)
First, calculate the dot product (u \\cdot v):
u \\cdot v = (-9)(-5) + (4)(9) = 45 + 36 = 81
Next, calculate the magnitudes of u and v:
|u| = \\sqrt{(-9)^2 + (4)^2} = \\sqrt{81 + 16} = \\sqrt{97}
|v| = \\sqrt{(-5)^2 + (9)^2} = \\sqrt{25 + 81} = \\sqrt{106}
Now, compute the cos(θ):
cos(θ) = 81 / (\\sqrt{97} \\cdot \\sqrt{106})
Use a calculator to find θ:
θ = cos^{-1}(81 / (\\sqrt{97} \\cdot \\sqrt{106}))
And θ is approximately 37 degrees, which makes option 3) 37 deg the correct answer.
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