find the angle between u=7i+2j and v=-4j
u is in first quadrant
tan of angle above x axis is 2/7
v is on -y axis
so the angle between them is
90 + tan^-1(2/7)
or, recall that
u dot v = |u| * |v| * cos(theta)
To find the angle between two vectors, you can use the dot product formula. The dot product of two vectors u and v is given by the formula:
u · v = |u| |v| cos(θ)
where |u| and |v| are the magnitudes (or lengths) of the vectors, and θ is the angle between them.
First, let's find the magnitudes of the vectors u and v:
|u| = sqrt((7^2) + (2^2)) = sqrt(49 + 4) = sqrt(53)
|v| = sqrt((-4)^2) = sqrt(16) = 4
Next, calculate the dot product of u and v:
u · v = (7)(0) + (2)(-4) = 0 - 8 = -8
Now, substitute the values into the dot product formula:
-8 = sqrt(53) * 4 * cos(θ)
To solve for θ, divide both sides of the equation by (sqrt(53) * 4):
-8 / (sqrt(53) * 4) = cos(θ)
-2 / (sqrt(53)) = cos(θ)
Using the inverse cosine function (cos^(-1)), we find:
θ = cos^(-1)(-2 / sqrt(53))
Calculating this using a calculator gives:
θ ≈ 109.47 degrees
Therefore, the angle between u and v is approximately 109.47 degrees.