CARTESIAN COMPONENTS OF VECTORS
The diagram shows a two-dimensional coordinate frame.
Any point P in the plane of the figure can be defined in terms of its x and y coordinates.
A unit vector pointing in the positive direction of the x-axis is denoted by i.
Any vector in the direction of the x-axis will be a multiple of i.
A vector of length l in the direction of the x-axis can be written li.
(All these vectors are multiples of i.)
(All these vectors are
multiples of j.)
A unit vector pointing in the positive direction of the y-axis is denoted by j.
Any vector in the direction of the y-axis will be a multiple of j.
A vector of length l in the direction of the y-axis can be written lj.
i represents a unit vector in the direction of the positive x-axis.
j represents a unit vector in the direction of the positive y-axis.
Draw the vectors 5i and 4j. Use your diagram and the triangle law of addition to add these two vectors together.
Any vector in the xy plane can be expressed in the form
r = ai + bj
The numbers a and b are called the components of r in the
x and y directions.
a) Draw an xy plane and show the vectors p = 2i + 3j, and q = 5i + j.
b) Express p and q using column vector notation.
c) Show the sum p + q.
d) Express the resultant p + q in terms of i and j.
If a = 9i + 7j and b = 8i + 3j, find:
a) a + b
b) a − b
The position vector of P with coordinates (a, b) is:
r = OP = ai + bj
State the position vectors of the points with coordinates:
a) P(2, 4)
b) Q(−1, 5)