Final answer:
To calculate the angle theta between vectors u and v, find the dot product, calculate the magnitudes of the vectors, and then use the cosine formula theta = cos^-1(u ⋅ v / (|u| |v|)). Finally, convert theta from radians to degrees and round to two decimal places.
Step-by-step explanation:
To find the angle theta between the vectors u and v, we use the dot product formula and the definition of the dot product in terms of vector components and the cosine of the angle between them. The vectors given are u = (0, -4) and v = (5, 3). The modified dot product formula according to the student's question is u ⋅ v = u₁v₁+ 3u₂v₂, which requires calculating the individual products of the corresponding components of u and v.
Let's calculate the dot product first:
- u ⋅ v = (0)(5) + 3(-4)(3) = 0 - 36 = -36
Now we need to find the magnitudes of the vectors:
- |u| = sqrt(0^2 + (-4)^2) = 4
- |v| = sqrt(5^2 + 3^2) = sqrt(25 + 9) = sqrt(34)
Using the dot product and magnitudes, we can find the angle theta using the following cosine formula:
cos(theta) = (u ⋅ v) / (|u| |v|)
cos(theta) = (-36) / (4 sqrt(34))
theta = cos^-1 (-36 / (4 sqrt(34)))
Calculating this will give us the angle in radians, which can then be converted to degrees. Remember to use inverse cosine (or cos^-1) to find the angle. Once calculated, round the decimal value of theta to two decimal places as requested.