Lesson 15 – Subtracting Vectors
 Vector subtraction
 Adding the opposite
 Application of triangle and parallelogram laws of vectors
Subtraction of vectors
In arithmetic, subtraction is the reverse operation of addition. When you have a question
such as 8 – 2 equals 6. The number 6 can be added to 2 to get 8.
With this understanding, the same principal is used with subtraction of vectors.
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When subtracting a - b , it is being asked, what vector added to b gives the sum a .
Vector Subtraction
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Let a and b be any two vectors. Either of the two methods shown below can be used to
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find a - b .
1. Identify head and tail:
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Arrange a and b tail to tail. Then a - b is the vector from the head of b to the head of a .
2. Add the opposite:
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a - b is the sum of a and the opposite of b . a - b = a +(- b )
Example 1
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Given the vectors u and v , draw the vector u - v .
a.
b.
Solution
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a. For this one because the vectors u and v both originate from the same point then the
identify the head and tail method should be used.
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Head of v to the head of u .
b. For this case since the vectors are consecutive the add the opposite method should be
used.
Example 2
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ABCD is a square. Express the difference of AC  BC as a single vector.
Solution
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AC and BC do not have the same tail.
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Since BC = AD , then BC can be replaced with AD .
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AC  BC  AC  AD
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 DC
Homework Questions
1. The diagram below shows three congruent equilateral triangles. Express each
difference as a single vector.
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a. BA  BC
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c. CE  AE
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b. BA  BD
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d. AE  ED
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2. Copy each set of vectors and draw u  v .
a.
b.
c.
3. ABCD is a rectangle. Express each vector as the difference of two other vectors. It may
be possible to do this in more than one way.
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a. BC
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b. DA
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c. BC
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d. CD
Part II
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4. TUVWXY is a regular hexagon. Determine TU  UV VW  WX  XY  YT
Part III
1. The diagram below shows two squares. Express each difference as a single vector.
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a. DB  DE
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c. AC  BD
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b. BE  BA
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2. Copy each set of vectors and draw u  v .
a.
b.
c.
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d. AE  ED
Part IV
3.
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In parallelogram EFGH, EF = u and FG = v . State a single vector equal to each of
the following.
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a. u  v
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c.  u  v
b. u  v
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d. v  u
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4. The diagram below shows a cube, where AB = u , AD = v and AE = w . Determine a
single vector equivalent to each of the following.
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a. u  v  w
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b. u  v  w
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c. u  v  w
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d. u  v  w