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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 point P in 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 l in 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 l in 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 thexy plane 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 anxy plane 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+ 7jandb= 8i+ 3j, find:a)a+b
b)ab
Example
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Key Point
The position vector ofP with 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|.
Whenr= ai+ bjthe modulus can be obtained using
Pythagoras theorem. Ifris the position vector of point P
then the modulus is clearly the distance ofP from 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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