For AI agents: use /llms.txt for the Nakafa content index.
How to find the matrix associated with a dilation operation? Recall that a point (x,y) is mapped by a dilation with a factor k=0 and center to .
Suppose the matrix we are looking for is (rtsu).
Find r,s,t,u such that it satisfies
From the matrix multiplication on the left side, we get:
By equating the corresponding components:
- First row: rx+sy=kx. For this equation to hold for all x and y, the coefficients of x must be equal, and the coefficients of y must be equal. Thus, r=k and s=0.
- Second row: tx+uy=ky. Similarly, t=0 and u=k.
The matrix associated with a dilation by a factor k=0 with respect to the origin O(0,0) is
A point (x,y) dilated by a factor k=0 and center (a,b) will be mapped to (k(x−a)+a,k(y−b)+b).
Find the combination of matrix operations on the position vector (x−ay−b) such that the result is (k(x−a)+ak(y−b)+b).
The matrix operation associated with a dilation by a factor k=0 with respect to the point (a,b) is
or it can also be written as:
Determine the image of point A(2,4) transformed by a dilation with a factor of 2 with respect to the center point P(1,1)!
Given x=2,y=4,k=2,a=1,b=1.
Thus, the image of point A(2,4) is A′(3,7).
- Find the coordinates of the image of the point (a,b) under the dilation [O,3]!
- Determine the matrix corresponding to a dilation with a scale factor of −2 and centered at O(0,0).
- A point B(−1,5) is dilated with center P(2,−3) and scale factor k=21. Determine the coordinates of the image of point !
- A triangle KLM with vertices K(1,1), L(5,1), and M(3,4) is dilated with center and scale factor . Draw the original triangle and its image, then determine the coordinates of the image vertices!
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The dilation [O,3] means the center of dilation is O(0,0) and the scale factor is k=3.
Let the point be Q(a,b).
Then x=a,y=b,k=3.
Thus, the coordinates of the image of point (a,b) are (3a,3b).
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Scale factor k=−2, center O(0,0).
The dilation matrix is:
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Point B(−1,5), center P(2,−3), scale factor k=21.
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Triangle KLM with K(1,1), L(5,1), M(3,4).
x=−1,y=5,a=2,b=−3,k=21.
The coordinates of the image of point B are B′(21,1).
Center O(0,0), k=2.
Image of point K(1,1):
Image of point L(5,1):
Image of point M(3,4):
Triangle
KLM is dilated to become triangle
K′L′M′. The center of dilation is
O(0,0).
