Course:MATH200/Vectors

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Vectors

Points

Let n be a non-negative integer and let R denote the set of real numbers. A point in Rn is an n-tuple (x1, x2,...,xn) of real numbers. The set of all points P in Rn is sometimes called affine n-space or n-dimensional Euclidean space.

Affine 2-space is also known as the Cartesian plane. Points in the Cartesian plane are simply ordered pairs (x,y) of real numbers.

Affine 3-space is also called Cartesian 3-space . Points in Cartesian 3-space are ordered triples (x,y,z) of real numbers.

Coordinates. If v =(v1,..., vn) is a point in Rn, then the real numbers v1,..., vn are called the coordinates of v. The number v1 is called the first coordinate, v2 is called the second coordinate and so on. If n=2, the first coordinate is normally called the x-coordinate and the second coordinate is normally called the y-coordinate. If n=3 the first coordinate is called the x-coordinate, the second is called the y-coordinate and the last coordinate is called the z-coordinate. So, for example, if v = (-1,5,7), then -1 is the x-coordinate, 5 is the y-coordinate and 7 is the z-coordinate.

Vectors

Like points, vectors in Rn are also ordered n-tuples ⟨x1, x2, ...,xn⟩. We write Rn for the set of all such vectors.

The only difference between points and vectors is how we visualize them. Points are visualized as dots and vectors are visualized as arrows. For example, the point (2,3) in the Cartesian plane is visualized as a dot 2 units to the right of the origin and 3 units above the origin. The vector ⟨2, 3⟩ can be visualized in infinitely many ways depending on where you start the arrow. The simplest way would be as an arrow starting at the origin in the Cartesian plane and ending at the point (2,3). However, we could also visualize ⟨2, 3⟩ as an arrow starting at (1,5) and ending at (3,8). More generally we can pick any point P=(a,b) in the Cartesian plane and visualize ⟨2, 3⟩ as the arrow starting at P and ending at (a+2,b+3).

If P=(p1, ...,pn) and Q=(q1, ...,qn) are two points in Rn then the vector from P to Q is the vector = ⟨q1-p1,...,qn-pn⟩. This is nothing but the vector starting at P and ending at Q. The position vector of P is the vector ⟨p1, ...,pn⟩. This is simply the vector from the origin to P.

Basic operations with Vectors

There are two basic operations with vectors: addition of vectors and scalar multiplication.

Addition Suppose v = ⟨ v1, ...,vn⟩ and w = ⟨ w1, ...,wn⟩ are two vectors in Rn. Then v + w :=⟨ v1 + w1, ...,vn + wn⟩. For example, ⟨ 1,2⟩ + ⟨ -3, 6⟩ = ⟨ -2, 7⟩. The vector v + w is called the sum of the two vectors v and w and the process computing the sum is known as vector addition.

Scalar Multiplication Suppose v = ⟨ v1, ...,vn⟩ is a vector in Rn and a is a real number. Then the product of a with v is the vector av:=⟨av1, ...,avn⟩. For example, if a=5 and v= ⟨ 1,2⟩ then 5v=5⟨ 1,2⟩ = ⟨5,10⟩.

The zero vector and the negative of a vector The zero vector in Rn is the vector 0=(0,0,...,0). It is used so often that it is convenient to have a special name for it. If v = (v1,...,vn) is a vector in Rn then, then negative of v is the vector -v:=(-v1,...,-vn).

Properties of adddition and scalar multiplication

Let u, v and w denote vectors in Rn and let a, b denote scalars in R. Then

  1. Vector addition is commutative: u + v = v + u.
  2. Vector addition is associative: u + (v + w)=( u + v) + w.
  3. u + 0 = u.
  4. u + (-u)=0.
  5. (ab)u = a(bu).
  6. 1u = u.
  7. Scalar multiplication is distributive: a(u + v )= au + av.
  8. (a+b)u = au + bu.