Question

Write the new coordinates after the triangle is rotated 90 degrees clockwise.

Answer 1

I am going to write some equations please wait a moment.

To rotate the triangle we use the rotation matix that rotates counterclockwise, but since we want to rotate clockwise we change the sing of the angle and 90 becomes -90:

`\begin{bmatrix}{\cos \text{ (}\theta)} & -\sin (\theta){} & {} \\ {\sin (\theta)} & {\cos (\theta)} & {} \\ {} & & {}\end{bmatrix}=\text{ }\begin{bmatrix}{\text{cos(-}90}) & {-\sin (-90)} & {} \\ {\sin (-90}) & {\cos (-90)} & \\ {} & {} & \end{bmatrix}=\text{ }\begin{bmatrix}{0} & {1} & \\ {-1} & {0} & {} \\ {} & {} & {}\end{bmatrix}`

Now, we put the points given in its vertical form

`\begin{gathered} A=\text{ }\begin{bmatrix}{2} & {} & {} \\ {10} & & {} \\ {} & {} & {}\end{bmatrix} \\ B=\text{ }\begin{bmatrix}{3} & {} & {} \\ {-4} & & {} \\ {} & {} & {}\end{bmatrix} \\ C=\text{ }\begin{bmatrix}{-3} & {} & {} \\ {1} & & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}`

Now we multiply the rotation matrix R with the points A, B, C and we will get the new coordinates A', B' and C' of the new rotated triangle

`\begin{gathered} A^(\prime)=R\cdot A=\text{ }\begin{bmatrix}{10} & {} & {} \\ {-2} & & {} \\ {} & {} & {}\end{bmatrix}\text{ } \\ B^(\prime)=R\cdot B=\text{ }\begin{bmatrix}{-4} & {} & {} \\ {-3} & & {} \\ {} & {} & {}\end{bmatrix} \\ C^(\prime)=R\cdot C=\text{ }\begin{bmatrix}{1} & {} & {} \\ {3} & & {} \\ {} & {} & {}\end{bmatrix} \end{gathered}`