Two-dimensional Coordinate frames P(x.y) 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.
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Y (All these vectors are multiples of i.) 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 I in the direction of the x-axis can be written li.
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Y (All these vectors are multiples of j.) 25 4j 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 I in the direction of the y-axis can be written Ij.
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Key Point 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.
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Example Draw the vectors 51 and 4j. Use your diagram and the triangle law of addition to add these two vectors together.
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Any vector in the xy plane can be expressed in the form The numbers a and b are called the components of in the x and y directions. 24 ai
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Example a) b) c) d) y Draw an xy plane and show the vectors 2 = 21 + 3j, and q = 51 + j. Express 2 and q using column vector notation. Show the sump + q. Express the resultant 2 + q in terms of i and j. 7 4j
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Example If a = 9i + 7j and b = 8i +3j, find:
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Key Point The position vector of P with coordinates (a, b) is: = OP = ai + bj P(a.b)
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Example State the position vectors of the points with coordinates: a) b) c) d) S(8, —4)
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Example Sketch the position vectors: y
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The modulus of any vector r is equal to its length. As we have noted earlier the modulus of r is usually denoted by Irl. When r = ai + bj the modulus can be obtained using Pythagoras' theorem. If r is the position vector of point P then the modulus is clearly the distance of P from the origin.
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Key Point if r = ai + bj then Ir
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Example Point A has coordinates (3, 5). Point B has coordinates (7, 8). a) Depict these points on a diagram. b) State the position vectors of A and B. c) Find an expression for AB. d) Find IABI.
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