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Vectors in geometry
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Vectors prove geometric facts: that lines are parallel, that points lie on a straight line, or exactly where one point divides a line.
Parallel vectors
Two vectors are parallel if one is a multiple of the other.
6i + 9j = 3(2i + 3j), so these are parallel.
a = 3i + 2j and ki + 8j parallel: k3 = 82, so k = 12.
6i + 9j = 3(2i + 3j), so these are parallel.
a = 3i + 2j and ki + 8j parallel: k3 = 82, so k = 12.
Collinear points
A, B and C are collinear (on one straight line) if AB and BC are parallel and share the point B.
AB = 2i + 3j and BC = 6i + 9j = 3AB, so A, B and C are collinear.
AB = 2i + 3j and BC = 6i + 9j = 3AB, so A, B and C are collinear.
Dividing a line in a ratio
If P divides AB in the ratio m : n, then
OP = n(m + n)a + m(m + n)b
The midpoint M: OM = 12(a + b)
P divides AB in the ratio 2 : 1: OP = 13a + 23b
OP = n(m + n)a + m(m + n)b
The midpoint M: OM = 12(a + b)
P divides AB in the ratio 2 : 1: OP = 13a + 23b
A(2, 1) and B(8, 13). P divides AB in the ratio 1 : 2. Find P.
- AB = 6i + 12j
- AP = 13AB = 2i + 4j
- OP = (2 + 2)i + (1 + 4)j
Answer: P is (4, 5)
Parallel means one vector is a multiple of the other. Collinear means parallel with a common point. P dividing AB in ratio m : n: OP = a + m(m + n)(b − a).
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