a) Suppose Graham's scan executes n points, where n >= 3. Prove that, at the end of the S consists of, from bottom to top, exactly the vertices of convex hull in counter-clockwise order. program, the

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Given vectors :u = (5,2) ; v = (-2,5) w = (0,3) ; q = (10,4)4.1 Calculate the following dot products:u.v ; (u.v).w ; u.(3w) , u.(w-v)4.2 Calculate‖?‖ ; d(u,v) ; ‖? − ?‖24.3 u and v are they orthogonal4.4 u and w are they orthogonal4.5 find c real number that satisfy q = c.u4.6 Deduce that q and u are parallel4.7 Normalize vector w

Given a Triangle of three vertices A(2,3), B(4,2), C(3,0), Apply the shearing to the Triangle with SH=4

Submit solutions to the following problems in a single pdf document preferably completed using LaTeX. It is likely that you will need to use programming (Python) or a matrix calculator to solve most of the problems. You do not need to submit the code, but you should include explanations of all your conclusions. 3. Consider the cat polygon with vertices at (0, 0), (3, 3), (3, 6), (2, 7), (1, 6), (-1, 6), (-2, 7), (-3, 6), and (-3, 3). a. Write a single matrix that will make the cat's face twice as tall, half as wide, and rotate anti- clockwise 45 degrees. What is the determinant of the transformation matrix? b. Calculate the area of the cat's face prior to transformation and after the transformation. How does the change in area relate to the determinant of the transformation matrix?

Recall that the dot product of two vectors (1d matrices) produces a scalar value. The dot product is slightly confusing as the scalar value produced can have an arbitrary meaning that simply represents the mathematical operations of multiplying and summing. In other words, the dot product can simply represent the linear projection of vector onto the number line. This interpretation will be used repeatedly throughout machine learning as our main goal will be to take some features dotted with some weights/parameters. Once again, this can be though of as projecting our features onto a number line where the projection acts as our prediction! Keep this idea in mind as it might not make complete sense as of yet. It also important to know that the dot product can additionally take on other meanings such as a geometric meaning which represents how similar any two vectors are when projected onto one another. Meaning, how much one vector points in the direction of another. Given the following…

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