Vectors - Parallel, Same Line, Ratios
This lesson covers:
- How to use vectors to check if a line is straight
- Using ratios to calculate line segment lengths
- Using vector ratios to determine the size and direction of vectors
Vectors along a straight line
Vectors lie along a straight line if they are scalar multiples of each other.
This confirms they share the same direction.
AB = a
BC = 2a
ABBC = a2a=2
BC is a multiple of AB, therefore the line ABC is straight. We say the vectors AB and BC are co-linear.
Worked example 1: Determining linearity
Determine if the line ABC is straight, given the following vectors:
AB = a
BD = −b
DC = 2a + b
Determine if the two vectors are co-linear.
a = (24) and b = (1612)
The two vectors are co-linear.
The two vectors are not co-linear.
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Vector questions involving ratios
Ratios can be used alongside vectors to determine the length and direction of segments along a line.
ABCD is a parallelogram.
AB and DC are parallel.
AD and BC are also parallel.
The line DC is split into DE and EC according to the ratio 4:1
Find the value of AE, given the following vectors:
AB=DC= a
BC=AD= b
ACE is a triangle.
B is the midpoint of AC.
D is the midpoint of AE.
AB = a
AD = b
Determine if CE is parallel to BD.
CE is not parallel to BD.
CE is parallel to BD.
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ABC is a triangle.
Point P lies on the line AB such that AP:PB is in the ratio 5:1.
AC=a and BC=b
Determine the value of CP in terms of a and b.
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