Lesson 16 – Scalar Multiplication of Vectors
 Vector multiplication
 Distributive law and vectors
 Linear combinations of vectors
Scalar multiplication of vectors
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When you add a to a you have a vector 2 a in length.
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The vector a is now twice as long or in other words has been multiplied by a scalar of 2.
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2( a )=2 a
Scalar Multiplication
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Let v be any vector and let k be a scalar. Then k v is a vector that is Ikl times as
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long as v .
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If k  0, k v has the same direction as v .
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If k  0, k v has the opposite direction as v .
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If k  0, k v is the zero vector v .
Scalar multiplication is also distributive.
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m( u  v ) = m u + m v
Example 1
In rectangle EFGH, X and Y are the midpoints of EF and EH respectively.
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If EX  u and EY  v , express each vector in terms of u and/or v .
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a. EF
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c. YG
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b. HE
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d. XG
Solution
a.
b.
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EF  2 EX
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HE  2 EY
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 2u
c.
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 2 v
d.
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XG  XY  YG
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YG  YH  HG
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 u v  v  2u
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 v  2u
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 u 2v
Linear Combination of Vectors
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If a and b are non-zero, non-collinear vectors, then any vector OP in the plane containing
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a and b can be expressed as a linear combination of a and b .
Example 2
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Draw any two non-collinear vectors u and v . Then draw each vector on the same diagram.
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a. w  2 u  4 v
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b. z  3 u  v
Solution
a.
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Draw u and v with a common tail, O.
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Draw the line l containing u .
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Locate point P on l such that OP  2 u . Through P, draw a line parallel to v . Locate point
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Q on this line such that PQ  4 v . Then w  OQ .
b.
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Locate point R on l such that OR  3 u . Through R, draw a line parallel to v . Locate point
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Q on this line such that RS   v . Then z  OS .
Homework Questions
1. The diagram below shows that segment AB and EH are parallel. Express each vector as
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a scalar multiple of u .
B
A
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b. EG
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f. FE
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a. EF
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e. FH
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c. EH
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g. GE
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d. FG
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h. HE
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2. In rectangle ABCD, E is the midpoint of AB, AE  u , and AD  v . Express each vector
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in terms of u and/or v .
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b. AC
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a. AB
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c. CE
3. Draw a diagram like the one below. Then draw each of the following on your diagram.
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a. 2 u
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
b.
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3 u 2v
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e.  u  3 v
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1
v c. 3 u  2 v
2
d.
Part II
1. The diagram below shows that segment AB and FH are parallel. Express each vector as
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a scalar multiple of u .
B
A
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a. EF
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e. FH
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b. EG
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f. FE
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c. EH
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g. GE
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d. FG
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h. HE
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2. Suppose EG =3 EF . Draw diagrams to support your answers to each question.
a. What conclusions can you make about line segments EG and EF?
b. What conclusions can you make about points E, F, and G?
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3. Use the diagram below to express each of the given vectors in terms of OP and OQ .
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a. OR b. OU
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c. OW
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d. OS
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e. OA f. OY
Part III
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4. Use the diagram below to express each vector as a linear combination of a and b .
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b. OD
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a. OC
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c. OE
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d. OF
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5. Draw any two non-collinear vectors u and v tail to tail. Then draw each of the
following on the same diagram.
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a. 3 u  2 v
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b.  3 u  4 v
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c. 2 u  3 v