Find the angle between u=(8,-3) and V=(-3,-8). Round to the nearest tenth of a degree
It's 45 degrees
u(8,-3), v(-3,-8).
Tan A = (-8-(-3))/(-3-8) = -5/-11 = 0.45455, A = 24.4o. S. of W.
u.v = -24+24 = 0
so u and v are perpendicular
Where did Henry go wrong? 5/11 is the slope of the line between the two points. It is not the tangent of the angle between the two vectors. For that, you need
tanθ = tan(arctan(-8/-3)-arctan(-3/8)) = tan(pi/2)
θ = pi/2
Also note that the two slopes are negative reciprocals:
-3/8 * 8/3 = -1
To find the angle between two vectors, you can use the dot product formula:
u · v = |u| |v| cos(theta)
Here, u · v represents the dot product of vectors u and v, |u| and |v| represent the magnitudes of vectors u and v, and theta represents the angle between them.
First, let's calculate the magnitudes of the vectors:
|u| = sqrt(8^2 + (-3)^2) = sqrt(64 + 9) = sqrt(73)
|v| = sqrt((-3)^2 + (-8)^2) = sqrt(9 + 64) = sqrt(73)
Next, let's calculate the dot product of the two vectors:
u · v = (8 * (-3)) + (-3 * (-8)) = -24 + 24 = 0
Now, substitute the magnitudes and the dot product into the formula:
0 = sqrt(73) * sqrt(73) * cos(theta)
Simplifying further:
0 = 73 * cos(theta)
To find the angle theta, divide both sides by 73:
cos(theta) = 0
Since cos(theta) = 0 when theta = 90 degrees, the angle between the two vectors is 90 degrees.
So, the angle between u=(8,-3) and V=(-3,-8) is 90 degrees.