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Section3.1Vectors

Objectives
  1. Learn how to add and scalar-multiply vectors in R n , both algebraically and geometrically.
  2. Understand linear combinations geometrically.
  3. Pictures: vector addition, vector subtraction, linear combinations.
  4. Vocabulary words: vector, linear combination.

Subsection3.1.1Vectors in R n

We have been drawing points of R n as dots in the line, plane, space, etc. We can also draw them as arrows. Since we have two geometric interpretations in mind, in what follows we will call an ordered list of n real numbers an element of R n .

Points and Vectors

A point is an element of R n , drawn as a point (a dot).

thepoint ( 1,3 )

A vector is an element of R n , drawn as an arrow.

thevector A 13 B

The difference is purely psychological: points and vectors are both just lists of numbers.

When we think of an element of R n as a vector, we will usually write it vertically, like a matrix with one column:

v = E 13 F .

We will also write 0 for the zero vector.

Why make the distinction between points and vectors? A vector need not start at the origin: it can be located anywhere! In other words, an arrow is determined by its length and its direction, not by its location. For instance, these arrows all represent the vector E 12 F .

Unless otherwise specified, we will assume that all vectors start at the origin.

Vectors makes sense in the real world: many physical quantities, such as velocity, are represented as vectors. But it makes more sense to think of the velocity of a car as being located at the car.

Note

Another way to think about a vector is as a difference between two points, or the arrow from one point to another. For instance, A 12 B is the arrow from ( 1,1 ) to ( 2,3 ) .

( 1,1 ) ( 2,3 ) A 12 B

Subsection3.1.2Vector Algebra and Geometry

Here we learn how to add vectors together and how to multiply vectors by numbers, both algebraically and geometrically.

Vector addition and scalar multiplication

  • We can add two vectors together:
    C abc D + C xyz D = C a + xb + yc + z D .
  • We can multiply, or scale, a vector by a real number c :
    c C xyz D = C c · xc · yc · z D .
    We call c a scalar to distinguish it from a vector. If v is a vector and c is a scalar, then cv is called a scalar multiple of v .

Addition and scalar multiplication work in the same way for vectors of length n .

The Parallelogram Law for Vector Addition

Geometrically, the sum of two vectors v , w is obtained as follows: place the tail of w at the head of v . Then v + w is the vector whose tail is the tail of v and whose head is the head of w . Doing this both ways creates a parallelogram. For example,

E 13 F + E 42 F = E 55 F .

Why? The width of v + w is the sum of the widths, and likewise with the heights.

v w w v v + w 5 = 1 + 4 = 4 + 1 5 = 2 + 3 = 3 + 2
Vector Subtraction

Geometrically, the difference of two vectors v , w is obtained as follows: place the tail of v and w at the same point. Then v w is the vector from the head of w to the head of v . For example,

E 14 F E 42 F = E 32 F .

Why? If you add v w to w , you get v .

v w v w
Scalar Multiplication

A scalar multiple of a vector v has the same (or opposite) direction, but a different length. For instance, 2 v is the vector in the direction of v but twice as long, and 1 2 v is the vector in the opposite direction of v , but half as long. Note that the set of all scalar multiples of a (nonzero) vector v is a line.

Somemultiplesof v . v 2 v 1 2 v 0 v Allmultiplesof v .

Subsection3.1.3Linear Combinations

We can add and scalar-multiply vectors in the same equation.

Definition

Let c 1 , c 2 ,..., c p be scalars, and let v 1 , v 2 ,..., v p be vectors in R n . The vector in R n

c 1 v 1 + c 2 v 2 + ··· + c p v p

is called a linear combination of the vectors v 1 , v 2 ,..., v p , with weights or coefficients c 1 , c 2 ,..., c p .

Geometrically, a linear combination is obtained by stretching / shrinking the vectors v 1 , v 2 ,..., v p according to the coefficients, then adding them together using the parallelogram law.

Figure17Linear combinations of two vectors in R 2 : move the sliders to change the coefficients of v 1 and v 2 . Note that any vector on the plane can be obtained as a linear combination of v 1 , v 2 with suitable coefficients.
Example(Linear Combinations of a Single Vector)

A linear combination of a single vector v = A 12 B is just a scalar multiple of v . So some examples include

v = E 12 F ,3 2 v = E 3 / 23 F , 1 2 v = E 1 / 2 1 F ,...

The set of all linear combinations is the line through v . (Unless v = 0, in which case any scalar multiple of v is again 0. )

v
Example(Linear Combinations of Collinear Vectors)

The set of all linear combinations of the vectors

v 1 = E 22 F and v 2 = E 1 1 F

is the line containing both vectors.

v 1 v 2

The difference between this and a previous example is that both vectors lie on the same line. Hence any scalar multiples of v 1 , v 2 lie on that line, as does their sum.