MHB Find Vector of Given Length in Particular Direction

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To find a vector of a specific length in the opposite direction of w = ai + bj, the vector can be expressed as a scalar multiple of -ai - bj. The required length of the new vector is 6, which means the scalar must be calculated as 6 divided by the length of w. This ensures that the resulting vector maintains the desired magnitude while pointing in the opposite direction. The discussion emphasizes the relationship between vector direction, magnitude, and scalar multiplication. Understanding these concepts is crucial for accurately determining vectors in specified orientations.
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Any vector "in the opposite direction as w= ai+ bj" is a number times -ai- bj. In order that the vector have length 6, that number must be 6 divided by the length of w.
 

Thread 'Area of Overlapping Squares'

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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