Final answer:
The image of the point (3,4) under a rotation of 45 degrees about the origin is (7√2/2, 7√2/2).
Step-by-step explanation:
To find the image of the point (3,4) under a rotation of 45 degrees about the origin, we can use the rotation matrix. The rotation matrix for a counterclockwise rotation of θ degrees about the origin is:
[ cosθ -sinθ ]
sinθ cosθ ]
Plugging in θ = 45 degrees, we get:
[ cos45 -sin45 ]
sin45 cos45 ]
Simplifying, we have:
[ √2/2 -√2/2 ]
√2/2 √2/2 ]
To apply this matrix to the point (3,4), we multiply:
[ √2/2 -√2/2 ] [ 3 ]
√2/2 √2/2 ] 4 ]
Simplifying, we get:
[ 7√2/2 ]
7√2/2 ]
So, the image of the point (3,4) under a rotation of 45 degrees about the origin is (7√2/2, 7√2/2).