Lect 5

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Lecture 5

VECTORS Be Superio r

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Introduction: scalar and vector quantities

Physical quantities can be divided into two main groups, scalar quantities and vector quantities. (a) A scalar quantity is defined completely by a single number with appropriate units (b) A vector quantity is defined completely when we know not only its magnitude (with units) but also the direction in which it operates

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Vector representation A vector quantity can be represented graphically by a line, drawn so that: (a) The length of the line denotes the magnitude of the quantity (b) The direction of the line (indicated by an arrowhead) denotes the direction in which the vector quantity acts.

The vector quantity ____ AB is referred to asAB or a

Programme 6: Vectors Vector representation Two equal vectors Types of vectors Addition of vectors The sum of a number of vectors

Programme 6: Vectors Vector representation Two equal vectors If two vectors, a and b, are said to be equal, they have the same magnitude and the same direction

Programme 6: Vectors Vector representation

If two vectors, a and b, have the same magnitude but opposite direction then a = −b

Programme 6: Vectors Vector representation Types of vectors ____

(a) A position vectorAB

occurs when the point A is fixed

(b) A line vector is such that it can slide along its line of action (c) A free vector is not restricted in any way. It is completely defined by its length and direction and can be drawn as any one of a set of equal length parallel lines

Programme 6: Vectors Vector representation Addition of vectors ____

The sum of two vectorsAB

____

____ ____

AB + BC = AC

or a + b = c

____

and BC

____

is defined as the single vector AC

Programme 6: Vectors Vector representation The sum of a number of vectors Draw the vectors as a chain.

____

____ ____

____

____

AB + BC + CD+ DE = AE ____

or a + b + c + d = AE

Programme 6: Vectors Vector representation The sum of a number of vectors If the ends of the chain coincide the sum is 0.

a+b +c+d = 0

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Components of a given vector

____

____ ____

____

____

____

AEby Just asAB + BC + CD+ DE can be replaced so any singlePT vector can be replaced by any number of component vectors so long as the form a chain beginning at P and ending at T.

____

PT = a + b + c + d

Programme 6: Vectors Components of a given vector Components of a vector in terms of unit vectors ____

The position vectorOP , denoted by r can be defined by its two components in the Ox and Oy directions as: r = a (along Ox) + b (along Oy) If we now define i and j to be unit vectors in the Ox and Oy directions respectively so that then:

a = ai and b = bj r = ai + bj

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Vectors in space

In three dimensions a vector can be defined in terms of its components in the three spatial direction Ox, Oy and Oz as: r = ai + bj + ck where k is a unit vector in the Oz direction The magnitude of r can then be found from Pythagoras’ theorem to be: r = a 2 +b 2 + c 2

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Direction cosines The direction of a vector in three dimensions is determined by the angles which the vector makes with the three axes of reference: r = ai + bj + ck so that a = cosα therefore a = r cosα r b = cos β therefore b = r cos β r c = cos γ therefore c = r cos γ r

Programme 6: Vectors Direction cosines

Since: a2 + b2 + c2 =r 2 then r 2 cos2 α + r 2 cos2 β + r 2 cos2 γ = r 2 then cos2 α + cos2 β + cos2 γ = 1

Programme 6: Vectors Direction cosines

Defining:

l = cosα m = cos β n = cos γ

then: l 2 + m2 + n 2 = 1 where [l, m, n] are called the direction cosines.

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Scalar product of two vectors

If a and b are two vectors, the scalar product of a and b is defined to be the scalar (number): ab cosθ where a and b are the magnitudes of the vectors and θ is the angle between them. The scalar product (dot product) is a.b = ab cosθ denoted by:

Programme 6: Vectors Scalar product of two vectors

If a and b are two parallel vectors, the scalar product of a and b is then: a.b = ab cos0 = ab Therefore, given: a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then:

a.b = a1b1 + a2b2 + a3b3

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Vector product of two vectors

The vector product (cross product) of a and b, denoted by: a×b is a vector with magnitude: ab sinθ a×b and a direction such that a, b and form a right-handed set.

Programme 6: Vectors Vector product of two vectors

If nˆ is a unit vector in the direction of: a×b then:

a × b = ab sinθ nˆ

Notice that: b × a = −a × b

Programme 6: Vectors Vector product of two vectors

Since the coordinate vectors are mutually perpendicular: i× j = k j× k = i k ×i = j and i × i = j× j = k × k = 0

Programme 6: Vectors Vector product of two vectors

So, given: a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then: a × b = (a2b3 − a3b2 )i − (a1b3 − a3b1) j + (a1b2 − a2b1)k That is: i j a × b = a1 a2 b1 b2

k a3 b3

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Angle between two vectors

Let a have direction cosines [l, m, n] and b have direction cosines [l′, m′, n′] ____ ____ OP′ OP Let and be unit vectors parallel to a and b respectively. ( PP′)2 = (l − l ′)2 + (m − m′)2 + (n − n′)2 = 2 − 2(ll ′ + mm′ + nn′) = 2 − 2cosθ by the cosine rule therefore

cosθ =ll ′ + mm′ + nn′

Programme 6: Vectors Introduction: scalar and vector quantities Vector representation Components of a given vector Vectors in space Direction cosines Scalar product of two vectors Vector product of two vectors Angle between two vectors Direction ratios

Programme 6: Vectors Direction ratios

Since

r = ai + bj + ck and a b c l = , m= , n= r r r

the components a, b and c are proportional to the direction cosines they are sometimes referred to as the direction ratios of the vector.

Programme 6: Vectors Learning outcomes Define a vector Represent a vector by a directed straight line Add vectors Write a vector in terms of component vectors Write a vector in terms of component unit vectors Set up a system for representing vectors Obtain the direction cosines of a vector Calculate the scalar product of two vectors Calculate the vector product of two vectors Determine the angle between two vectors Evaluate the direction ratios of a vector

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