Suppose you want the equation of a line in E2 that is parallel to v and passes t

Question

Suppose you want the equation of a line in E2 that is parallel to v and passes through point p. (see picture attached) then consider W={w "e" E2|w= alpha v} Use the Gram-Schmidth process to find a basis for the complimentary subspase W(perp) of W in E2. Use the basis of W and the basis of W(perp) to create an orthonormal basis for E2. It is easy to see that every point x on the line can be expressed as p+alpha v for some scalar alpha. Use this fact to show that x-proj W on x= proj W on p for every x on the line. Since x=proj W on x+proj W(perp) on x => proj W(perp) on x = proj W(perp) on p for every x on the line. If {u1 u2} is an orthonormal basis of E2 with u1 e W and u2 e W(perp) respectivly => => x=(x dot u1)u1+(x dot u2)u2=(x dot u1)u1+(p dot u2)u2 for all x on the line => x dot u2 = (p dot u2) for all x on the line. This is called the normal equation of the line. Use this procedure to write the normal equation of a line through point (2,3) and (5,7)

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the question is too confusing please make it a bit shorter and then re post thanks

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