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Find the angle \(\theta\) between \(\mathbf{u} = -9i + 4j\) and \(\mathbf{v} = -5i + 9j\).

1) 35.8 degrees
2) 38.2 degrees
3) 37 degrees
4) 39 degrees

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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