MATH 3510 Chapter Notes - Chapter 12: Unit Vector, John Wiley & Sons

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3 Mar 2019

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Find the distance between a point and a line in or, find the distance between two parallel planes in, find the distance between a point and a plane. In exercises 1 2, determine whether u and v are orthogonal vectors: (a) (b) (c) (d) Answer: (a) orthogonal (b) not orthogonal (c) not orthogonal (d) not orthogonal: (a) (b) (c) (d) In exercises 3 4, determine whether the vectors form an orthogonal set: (a) (b) (c) (d) Answer: (a) not an orthogonal set (b) orthogonal set (c) orthogonal set (d) not an orthogonal set: (a) (b) (c) (d, find a unit vector that is orthogonal to both and. Answer: (a) show that and are orthogonal vectors. (b) use the result in part (a) to find two vectors that are orthogonal to (c) find two unit vectors that are orthogonal to, do the points. Yes: repeat exercise 7 for the points.

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1. Let V be an inner product space. Prove the Pythagoreantheorem: if v and w E V are orthogonal then ||v+w||^2 =||v||^2 + ||w||^2. (Suggestion: start with ||v+w||^2 = (v+w,v+w).)

2. Let V=P2, the polynomials of degree less than/equal to 2 withcoefficients in R, and let (.,.): VxV -> R be the map

(p,q) = p(0)q(0) + p(1)q(1) + p(2)q(2) for any two polynomialsp(x),q(x) E V. So, for example,

(x+2, x^2-1) = (0+2)(0^2-1) + (1+2)(1^2 - 1) + (2+2)(2^2-1) =10.

a) Show that the function(.,.) defines an inner product onV.

b) Is the set A=(1,x,x^2) an orthogonal set? (with respect tothe inner product above.)

c) Is the set B=(1,x-1,3x^2-6x+1) an orthogonal set?

d) Calculate ||1||^2, and ||3x^2-6x+1||^2

e) Is the set B an orthonormal set?

f) Write f=x^2-6x+12 as a linear combination of the vectors inB.

MATH 3510 Chapter Notes - Chapter 12: Unit Vector, John Wiley & Sons

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