Question

Give a point (1,2) an age, geometric figure, what is the new point when the figure is rotated clockwise about the origin 180°?

Answer 1

To rotate a point `(1,2)` by 180° clockwise about the origin, we need to multiply its coordinates by the rotation matrix:

```
[ cos(θ) -sin(θ) ] [ 1 ] [ cos(θ)*1 - sin(θ)*2 ] [ -1 ]
[ sin(θ) cos(θ) ] * [ 2 ] = [ sin(θ)*1 + cos(θ)*2 ] = [ 2 ]
```

In this case, `θ` is 180°, so we have:

```
[ cos(180°) -sin(180°) ] [ 1 ] [ cos(180°)*1 - sin(180°)*2 ] [ -1 ]
[ sin(180°) cos(180°) ] * [ 2 ] = [ sin(180°)*1 + cos(180°)*2 ] = [ 2 ]
```

Evaluating the trigonometric functions for 180°, we get:

```
[ -1 ]
[ 2 ]
```

Therefore, the new point after rotating the figure clockwise about the origin 180° is `(-1, 2)`.