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).