Developing geometric insight – R 2 and R 3

4

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4.2.1 Vectors in R

²

The set R can be represented as points along a horizontal line, called a real-number line.

In order to represent pairs of real numbers, (a1, a2), we use a Cartesian plane, a plane with both a horizontal axis and a vertical axis, each axis being a copy of the

real-number line, and we markA = (a₁, a₂) as apoint in this plane. We associate this point with the vector a= (a1, a2)^T, as representing a displacement from the origin (the point (0,0)) to the point A. In this context, ais the position vector of the point A.

This displacement is illustrated by an arrow, or directed line segment, with initial point at the origin and terminal point at A.

x-

y 6

(0,0)

>

r(a₁, a₂) a₂

a₁ a

position vector, a

Even if a displacement does not begin at the origin, two displacements of the same length and the same direction are considered to be equal. So, for example, the two arrows below represent the same vector,v= (1,2)^T.

x-

y 6

(0,0)

displacement vectors, v

If an object is displaced from a point, say O, the origin, to a point P by the

displacement p, and then displaced from P to Q, by the displacement v, then the total displacement is given by the vector from O toQ, which is the position vector q. So we would expect vectors to satisfy q=p+v, both geometrically (in the sense of a

displacement) and algebraically (by the definition of vector addition). This is certainly true in general, as illustrated below.

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4.2. Developing geometric insight –R²andR³

- 6

(0,0)

:

>

*

p₂

p₁ q₂

q₁ p

v q

If v= (v1, v2)^T, then q1 =p1+v1 and q2 =v2+p2.

The order of displacements does not matter, (nor does the order of vector addition) so also q=v+p. For this reason the addition of vectors is said to follow the parallelogram law.

- 6

(0,0)

:

>

:

>

p

v p v

p+v=v+p.

From q=p+v, we have v=q−p. This is the displacement from P to Q. To help you determine in which direction the vector vpoints, think ofv=q−p as the vector which is added to the vector p in order to obtain the vector q.

If vrepresents a displacement, then 2v must represent a displacement in the same direction, but twice as far, and −v represents an equal displacement in the opposite direction. This interpretation is compatible with the definition of scalar multiplication.

Activity 4.4 Sketch the vector v= (1,2)^T in a coordinate system. Then sketch 2v and −v. Looking at the coordinates on your sketch, what are the components of 2v and −v?

We have stated that a vector has both a length and a direction. Given a vector

a= (a₁, a₂)^T, its length, denoted bykak, can be calculated using Pythagoras’ theorem applied to the right triangle shown below:

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x-

y 6

(0,0)

>

(a₁, a₂) a₂ a₁

a

So the length of ais the scalar quantity kak=

q

a²₁+a²₂.

The length of a vector can be expressed in terms of the inner product, kak=p

ha,ai,

simply because ha,ai=a²₁ +a²₂. A unit vector is a vector of length 1.

Example 4.2 Ifv= (1,2)^T, then kvk=√

1²+ 2² =√ 5.

The vector u=

√1 5,^√2

5

T

is a unit vector in the same direction asv.

Activity 4.5 Check this. Calculate the length of u.

The direction of a vector is essentially given by the components of the vector. If we have two vectors a and b which are (non-zero) scalar multiples, say

a=λb, λ∈R, (λ6= 0),

then a and b are parallel. If λ >0 then a and b have the same direction. If λ <0 then we say that a and b have opposite directions.

The zero vector, 0, has length 0 and has no direction. For any other vector, v6=0, there is one unit vector in the same direction as v, namely

u= 1 kvkv.

Activity 4.6 Write down a unit vector, u, which is parallel to the vectora= 4

3

. Then write down a vector, w, of length 2 which is in the opposite direction to a.

4.2.2 Inner product

The inner product in R² is closely linked with the geometric concepts of length and angle. If a= (a₁, a₂)^T, we have already seen that

kak² =ha,ai=a²₁+a²₂.

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4.2. Developing geometric insight –R²andR³

Leta, b be two vectors in R², and let θ denote the angle between them. (Note that angles are always measured in radians, not degrees, here. So, for example 45 degrees is π/4 radians.) By the angle between two vectors we shall always mean the angle, θ, such that 0≤θ ≤π. Ifθ < π, the vectors a, b, and c=b−a form a triangle, where cis the side opposite the angle θ, as, for example, in the figure below.

- J

J J

J J

J JJ^

>

a c=b−a

b θ

The law of cosines (which you may or may not know — don’t worry if you don’t) applied to this triangle gives us the important relationship stated in the following theorem.

Theorem 4.3 Let a,b ∈R² and let θ denote the angle between them. Then ha,bi=kak kbkcosθ .

Proof

The law of cosines states that c² =a²+b²−2abcosθ wherec=kb−ak, a=kak, b=kbk. That is,

kb−ak² =kak²+kbk²−2kak kbkcosθ (1) Expanding the inner product and using its properties, we have

kb−ak² =hb−a,b−ai=hb,bi+ha,ai −2ha,bi That is,

kb−ak² =kak²+kbk²−2ha,bi (2) Comparing equations (1) and (2) above, we conclude that

ha,bi=kak kbkcosθ .

This theorem has many geometrical consequences.

For example, we can use it to find the angle between two vectors by using cosθ = ha,bi

kak kbk.

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Example 4.3 Let

v= 1

2

, and w= 3

1

, and let θ be the angle between them. Then

cosθ= 5

√5√

10 = 1

√2, so that θ= π 4.

Since

ha,bi=kak kbkcosθ ,

and −1≤cosθ ≤1 for any real number θ, the maximum value of the inner product is ha,bi=kak kbk. This occurs precisely when cosθ= 1, that is, when θ= 0. In this case the vectors a and b are parallel and in the same direction. If they point in opposite directions, then θ =π and we haveha,bi=−kak kbk. The inner product will be

positive if and only if the angle between the vectors is acute, meaning that 0≤θ < ^π₂. It will be negative if the angle is obtuse, meaning that ^π₂ < θ ≤π.

The non-zero vectors a and b are orthogonal (or perpendicular or, sometimes, normal) when the angle between them is θ= ^π₂. Since cos(^π₂) = 0, this is precisely when their inner product is zero. We restate this important fact:

The vectors a and b are orthogonal if and only if ha,bi= 0.

4.2.3 Vectors in R

³

Everything we have said so far about the inner product and its geometric interpretation inR² extends to R³.

If a=



 a₁ a₂ a₃



, then kak= q

a²₁+a²₂+a²₃.

Activity 4.7 Show this. Sketch a position vector a= (a1, a2, a3)^T in R³. Drop a perpendicular to the xy-plane as in the figure below, and apply Pythagoras’ theorem twice to obtain the result.

-y

6

z

x

(a1, a2, a3)

(a1, a2,0)

1 PP

PP PP

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