A rectangular playground is 64 ft long and 48 ft wide. there is a light pole for evening play in one corner of the playground, with the light placed 18 ft high. Find |v|, the length of the vector formed from the top of the light pole to the opposite corner of the playground. If u is the vector formed by the opposite end of the playground and the base of the pole, find the angle between these two vectors using `cos theta = (u.v)/(|u||v|)`. Verify the result using right triangle trigonometry
A rectangular playground is 64 ft long and 48 ft wide. There is a light pole for evening play in one corner of the playground, with the light placed 18 ft high.
Let the corner diagonally opposite the one where the light pole is located have coordinates (0,0,0). The coordinates of the corner where the light pole is located is (64, 48, 0)
The coordinates of the top of the light pole is (64, 48, 18). The vector `vec v` is [64, 48, 18]. The magnitude of `vec v` is `sqrt(64^2 + 48^2 + 18^2)` = 82
The vector `vec u` is [64, 48, 0]. The magnitude of `vec u` is `sqrt(64^2 + 48^2+ 0^2)` = 80
The dot product of the vectors, `vec u @ vec v` = 64^2 + 48^2 = 6400 = `|vec u|*|vec v|*cos theta` where `theta` is the angle between the vectors.
`cos theta = 6400/(80*82) `
=> `theta = cos^-1(6400/(80*82))`
=> theta = 12.68 degrees
Considering a right triangle with hypotenuse, 82, and the other sides 80 and 18, the angle between the side with length 80 and the hypotenuse is `tan^-1(18/80)` = 12.68 which is the same as the value of `theta` derived earlier.
The angle between the vectors `vec u` and `vec v` is 12.68 degrees.