Lesson 14 – Adding Vectors
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Triangle law of vectors
Zero vector
Parallelogram law of vectors
Vector addition
Addition of vectors
The diagram below shows a point A being translated to point B then from that point B to a
point C. A single displacement from A to C is the same as previously described.
Three vectors together forming a triangle through addition is called the Triangle Law.
Triangle Law of Vector Addition
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Let a and b be any two vectors arranged head-to-tail. The sum, a + b , is the vector from
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the tail of a to the head of b .
Solution
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a. Arrange the vectors in order by placing the tail of b to the head of a .
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Then draw a vector from the tail of a to the head of b .
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This is vector a + b .
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b. Arrange the vectors in order by placing the tail of a to the head of b .
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Then draw a vector from the tail of b to the head of a .
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This is vector b + a .
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c. The vectors a + b and b + a have the same magnitude and direction. So, they are equal
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vectors. So a + b = b + a .
Solution
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a  b  c  d  AB  BC  CD  DE
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 AC  CD  DE
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 AD  DE
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 AE
Example 3
The diagram below shows a rectangular prism. Determine a vector equal to each sum.
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a. AE  HC
Solution
a.
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b. AD  AE  AB
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The sum of a and b is the vector with the same tail as a and b and with its head at the
opposite vertex of the parallelogram.
Example 4
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Draw u  v .
Solution
Homework Questions
1. Express each sum as a single vector.
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a. AB BC
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b. AC  CD
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c. (BC  CD) DA
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d. BC  (CD DA)
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e. CA AD DB
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f. BD DB
Part II
2. In the diagram below, ABCD and CEFG are parallelograms. Express each sum as a
single vector.
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a. HG HD
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b. HG HA
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c. FG FE
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d. CD HG
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3. Copy each set of vectors and draw u  v .
a.
b.
c.
4. Use a diagram to explain how each vector sum can be expressed as a single vector.
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a. AB  BC  CD
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b. PQ  RP
Part III
1. Express each sum as a single vector.
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a. PT  TQ
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b. QR  RU
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c. RV  VS
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d. PV  VS
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e. UQ QW  WV
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f. SW  WQ  QR
2. In the diagram below, ABC is an equilateral and D, E, F are the midpoints of its sides.
Express each sum as a single vector.
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a. AF  DB
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b. DE  DB
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e. AF  DE
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f. EC FD
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c. FA EB
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d. DA EC
Part IV
3. The diagram below shows a square-based pyramid. Determine each sum.
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a. KN  NR
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b. RS KR
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e. KN  RS
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f. KR  NM  SK
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c. MN  MS
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d. KM  NK
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4. Copy each set of vectors and draw u  v or u  v  w .
a.
b.
d.
e.
c.
5. Use a diagram to explain how each vector sum can be expressed as a single vector.
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a. AB  CA
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b. ST  US VU