n Section 5.1 - The Scalar Product in R

In this chapter, we extend the concept of the dot product to general vector spaces. We begin in n this section defining the scalar product on R . n Definition. Given x, y ∈ R we call the quantity

T x y = x1y1 + ··· + xnyn the scalar product. This is just the dot product, but in matrix multiplication form. We can also define the Euclidean length or norm of x as q T
1/2 2 2 kxk = x x = x1 + ··· + xn which leads to the Euclidean distance between x and y given by

p 2 2 kx − yk = (x1 − y1) + ··· + (xn − yn)

2 In R , we can compute the angle θ between two vectors geometrically and show the dot product formula xT y = kxkkyk cos θ. In this way, we generalize the definition of the the angle between n two vectors x, y ∈ R by xT y cos θ = kxkkyk

n We can also define scalar projection and vector projection in R based on our geometric 2 intuition in R :

xT y xT y xT y y Scalar Projection: α = and Vector Projection: p = y = y = α kyk yT y kyk2 kyk

Note. The vector projection is just the vector in the direction of y with length α.

n T Also generalizing to R , we say two vectors x, y are orthogonal if x y = 0 and we write x ⊥ y. n M309 Notes ©, R.G. Lynch, Texas A&M Section 5.1 - The Scalar Product in R Page 2 of 4 √ Example. If x = (3, 4, −2, 0)T and y = (−1, 7, 2, 8)T ,

xT y =

x u := = kxk

y v := = kyk

kx − yk =

cos θ =

α =

p =

n n Example. Show that the zero vector in R is orthogonal to any other vector x ∈ R .

Example. Find the value of a so that x = (1, 2, 3, 4, 5)T and y = (−1, 2, a, 3a, 7)T are orthogonal. n M309 Notes ©, R.G. Lynch, Texas A&M Section 5.1 - The Scalar Product in R Page 3 of 4 # – n If P and Q are two points in R , we denote the vector from P to Q by PQ.

Definition.# – If N is a nonzero vector and P0 is a fixed point, then the set of points P such that P0P is orthogonal to N forms a hyperplane Π in n-space. The vector N and the hyperplane Π 3 are said to be normal to each other. In R , a hyperplane is just a plane as you’re accustomed to, but hyperplane is the higher dimensional analog - it is defined by a linear equation in n variables instead of 3. That is, a point P = (p1, . . . , pn) will lie on Π if and only if # – T P0P N = 0.

3 T In the case of R , if P0 = (x0, y0, z0), P = (x, y, z) and N = (a, b, c) , then this equation is

a(x − x0) + b(y − y0) + c(z − z0) = 0.

You can now see how this would generalize from (x, y, z) to any number of variables.

Example. Find the equation of the plane through the point (2, −1, 3) and normal to N = (2, 3, 4)T .

Example. Find the equation of the plane passing through

P1 = (1, 1, 2),P2 = (2, 3, 3),P3 = (3, −3, 3) n M309 Notes ©, R.G. Lynch, Texas A&M Section 5.1 - The Scalar Product in R Page 4 of 4

Example. Find the distance from the point (2, 0, 0) to the plane x + 2y + 2z = 0.

More generally, the distance from a point P to the plane a(x − x0) + b(y − y0) + c(z − z0) = 0 is

T v N d = |scalar projection| = kNk # – where v = P0P .

n 2 2 2 Theorem (Pythagorean Law) If x, y ∈ R are orthogonal then kx + yk = kxk + kyk n T Theorem (Cauchy-Schwarz Inequality). If x, y ∈ R , then x y ≤ kxkkyk with equality if and only if x and y are linearly dependent.