Cartesian components_vector.ppt

7/27/2019 Cartesian components_vector.ppt

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CARTESIAN COMPONENTS

OF VECTORS


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Two-dimensional Coordinate frames

The diagram shows a two-dimensional coordinate frame.

Any pointPin the plane of the figure can be defined in

terms of itsx andy coordinates.


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A unit vector pointing in the positive direction of thex-

axis is denoted by i.

Any vector in the direction of thex-axis will be a

multiple ofi.

A vector of length lin the direction of thex-axis can be

written li.

(All these vectors are multiples ofi.)


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(All these vectors aremultiples ofj.)

A unit vector pointing in the positive direction of the

y-axis is denoted byj.

Any vector in the direction of they-axis will be amultiple ofj.

A vector of length lin the direction of they-axis can

be written lj.


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Key Point

i represents a unit vector in the direction

of the positivex-axis.

j represents a unit vector in the direction

of the positivey-axis.


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Example

Draw the vectors 5i and 4j. Use your diagramand the triangle law of addition to add these twovectors together.


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Any vector in thexyplane can be expressed in the formr= ai + bj

The numbers a and b are called the components ofrin the

x andy directions.


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a) Draw anxyplane and show the vectorsp = 2i + 3j,and q = 5i +j.

b) Expressp and q using column vector notation.

c) Show the sump + q.

d) Express the resultantp + q in terms ofi andj.

Example


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Ifa= 9i+ 7jand b= 8i+ 3j, find:a) a+ b

b) a b

Example


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Key Point

The position vector ofPwith coordinates (a, b) is:

r= OP= ai + bj


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State the position vectors of the points with coordinates:a) P(2, 4)

b) Q(1, 5)c) R(1,7)d) S(8,4)

Example


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Example

Sketch the position vectors:r= 3i + 4j,

r= 2i + 5j,r= 3i 2j.

1

2

3


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The modulus of any vectorris equal to its length. As wehave noted earlier the modulus ofris usually denoted by |r|.

When r= ai+ bjthe modulus can be obtained using

Pythagoras theorem. Ifris the position vector of pointP

then the modulus is clearly the distance ofPfrom the origin.


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Key Point

ifr= ai+ bjthen |r| = (a+ b)


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PointA has coordinates (3, 5). PointB has coordinates (7, 8).a) Depict these points on a diagram.

b) State the position vectors ofA andB.

c) Find an expression forAB.

d) Find |AB|.

Example


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Cartesian components_vector.ppt