what is the angle between D=5i-3j and E=2i+j
To find the angle between two vectors, you can use the dot product formula:
θ = arccos((D · E) / (|D| * |E|))
where D · E is the dot product of vectors D and E, and |D| and |E| are the magnitudes of vectors D and E, respectively.
First, let's calculate the dot product:
D · E = (5 * 2) + (-3 * 1) = 10 - 3 = 7
Then, let's calculate the magnitudes:
|D| = sqrt((5)^2 + (-3)^2) = sqrt(25 + 9) = sqrt(34)
|E| = sqrt((2)^2 + (1)^2) = sqrt(4 + 1) = sqrt(5)
Now, let's substitute these values into the formula:
θ = arccos(7 / (sqrt(34) * sqrt(5)))
θ ≈ arccos(7 / (5.83095 * 2.23607))
θ ≈ arccos(7 / 13.03349)
θ ≈ arccos(0.53672)
Using a calculator, we find that the angle, θ, is approximately 57.06 degrees.