270 rule represent a 270 rotation to the left which is very easy
(x; y) --> (x.cos45 + y.sin45; x.sin45 - y.cos45)
The effect of the rotation is the same as that of a 90 degree clockwise rotation. In matrix notation, it is equivalent to [post-]multiplication by the 2x2 matrix: { 0 1 } {-1 0 }
we swap the co-ordinates and give the new y co-ordinate the opposite sign.90 degrees clockwise(y, -x)
It is multiplication by the 2x2 matrix 0 1-1 0
rosting method rule method set-builder rotation
(x; y) --> (x.cos45 + y.sin45; x.sin45 - y.cos45)
A counterclockwise rotation of 270 degrees about the origin is equivalent to a clockwise rotation of 90 degrees. To apply this transformation to a point (x, y), you can use the rule: (x, y) transforms to (y, -x). This means that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate.
The rule for a rotation by 180° about the origin is (x,y)→(−x,−y) .
The effect of the rotation is the same as that of a 90 degree clockwise rotation. In matrix notation, it is equivalent to [post-]multiplication by the 2x2 matrix: { 0 1 } {-1 0 }
The rotation rule for a 180-degree counterclockwise rotation involves turning a point around the origin (0, 0) by half a circle. For any point (x, y), the new coordinates after this rotation become (-x, -y). This means that both the x and y coordinates are negated. For example, the point (3, 4) would rotate to (-3, -4).
To rotate a figure 180 degrees clockwise, you can achieve this by first reflecting the figure over the y-axis and then reflecting it over the x-axis. This double reflection effectively rotates the figure 180 degrees clockwise around the origin.
To rotate a figure 90 degrees clockwise around the origin on a coordinate grid, you can use the transformation rule: (x, y) becomes (y, -x). For the point (5, 5), applying this rule results in (5, -5). Therefore, after a 90-degree clockwise rotation, the new coordinates of the point are (5, -5).
we swap the co-ordinates and give the new y co-ordinate the opposite sign.90 degrees clockwise(y, -x)
It is multiplication by the 2x2 matrix 0 1-1 0
(x,y) to (x,-y). You would keep the x the same, but turn the y negative. This is actually the rule for a 90 degree counterclockwise rotation, but they're the same thing, they would go to the same coordinates.
First of all, if the rotation is 180 degrees then there is no difference clockwise and anti-clockwise so the inclusion of clockwise in the question is redundant. In terms of the coordinate plane, the signs of all coordinates are switched: from + to - and from - to +. So (2, 3) becomes (-2, -3), (-2, 3) becomes (2, -3), (2, -3) becomes (-2, 3) and (-2, -3) becomes (2, 3).
(x,y) to (x,-y). You would keep the x the same, but turn the y negative. This is actually the rule for a 90 degree counterclockwise rotation, but they're the same thing, they would go to the same coordinates.