This question aims to find the type of rule that would be applied to the trapezoid ABCD with a point A( 0, -4 ) to rotate it to 270° in the clockwise direction.
A quadrilateral having two sides parallel to each other is called a trapezoid. This four-sided figure is also called a trapezium. When we need to find the rotation of a point in the trapezoid, we use the rotation matrix. A transformation matrix rotated in such a way that all its elements get rotated in Euclidean space then it is called a rotation matrix.
The order of the rotation matrix is $ n \times n $ in the n-dimensional space. Similarly, a matrix in a 3-D space will have an order of $ 3 \times 3 $.
Expert Answer
The rotation of a point ( x, y ) in the clockwise direction along an angle $ \theta $ in the coordinate plane is given by the rotation matrix. The order of the rotation matrix is $ n \times n $ in the n-dimensional space.
\begin{bmatrix}
\cos \theta & \sin \theta \\
– \sin \theta & \cos \theta
\end{bmatrix}
By putting the value of the angle $ \theta = 270 ° $
\begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix}
The rotation of matrix rule is applied as:
\[ \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix} \begin{bmatrix}
0 & 4
\end{bmatrix} \]
By multiplying the matrix with 0 and 4:
\[ \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
0 \cos 270 + 4 \sin 270 \\
– 0 \sin 270 + 4 \cos 270
\end{bmatrix} \]
\[ \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
4 \ sin 270 \\
4 \ cos 270
\end{bmatrix} \]
Numerical Results
The rule to find the rotation of a trapezoid in the clockwise 270 ° is rotation rule that is given by:
$ \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
4 \ sin 270 \\
4 \ cos 270
\end{bmatrix} $
Example
Rotate the trapezoid having a point ( 0, -3) in the clockwise direction along the angle $ \theta $.
\begin{bmatrix}
\cos \theta & \sin \theta \\
– \sin \theta & \cos \theta
\end{bmatrix}
By putting the value of the angle $ \theta = 270 ° $
\begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix}
The rotation of matrix rule is applied as:
\[ \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix} \begin{bmatrix}
0 & 3
\end{bmatrix} \]
By multiplying the matrix with 0 and 3:
\[ \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
0 \cos 270 + 3 \sin 270 \\
– 0 \sin 270 + 3 \cos 270
\end{bmatrix} \]
\[ \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
3 \ sin 270 \\
3 \ cos 270
\end{bmatrix} \]
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