I(hat), J(hat, k(hat) and vectors

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The hat symbols in vector notation represent unit vectors, which have a magnitude of one. Specifically, i-hat, j-hat, and k-hat denote the unit vectors in the directions of the positive x, y, and z axes, respectively. These unit vectors are fundamental in physics and engineering for defining directions in three-dimensional space. Understanding these symbols is essential for vector operations and calculations. Mastery of i-hat, j-hat, and k-hat is crucial for anyone working with vectors.
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could somebody better explain to me what these hat symbols mean when used with vectors.
 
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They are unit vectors (i.e., magnitude 1) in the direction of positive x, positive y, and positive z, respectively.
 
Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?

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