Point A lies at (2, 3) on a coordinate system. If the coordinate axes are rotated 45°, what is its position on the new coordinate system?
To find the position of point A on the new coordinate system after rotating the coordinate axes by 45°, we can use a rotation matrix.
A rotation matrix is a matrix that describes the rotation of a coordinate system around an origin point. For a rotation of 45° counterclockwise, the rotation matrix is:
R = [[cosθ, -sinθ], [sinθ, cosθ]]
where θ is the angle of rotation in radians.
Since we are rotating the coordinates axes by 45°, we can substitute θ = π/4 radians into the rotation matrix.
R = [[cos(π/4), -sin(π/4)], [sin(π/4), cos(π/4)]]
= [[√2/2, -√2/2], [√2/2, √2/2]]
To find the position of point A on the new coordinate system, we multiply the rotation matrix by the coordinates of point A:
A' = R * A
Let's substitute the values for A(x, y) = (2, 3):
A' = [[√2/2, -√2/2], [√2/2, √2/2]] * [2, 3]
Multiplying the matrices, we get:
A' = [(√2/2 * 2) + (-√2/2 * 3), (√2/2 * 2) + (√2/2 * 3)]
= [-√2 - (3√2/2), √2 + (3√2/2)]
= [-√2 - (3√2/2), √2 + (3√2/2)]
Therefore, the position of point A on the new coordinate system after rotating the coordinate axes by 45° is approximately (-2.12, 5.12).