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Scalar Multiplication
Given a vector multiplied by a constant
all components are multiplied by the constant
Example
,3(3)>=<6,9> "3<2,3> = <3(2),3(3)>=<6,9>")
or
=6\vec{i}+9\vec{j} "3(2\vec{i}+3\vec{j})=6\vec{i}+9\vec{j}")
Vector Addition and Subtraction
Given two vectors
then
which has both of the following geometrical interpretations:
and
also,
which has the following geometrical interpretation:
Example
Find the sum and difference of the given vectors
Solution:
+(-7)> =<4,-9> "\vec{a}+\vec{b}=<3+1,(-2)+(-7)> =<4,-9>")
and
-(-7)> =<2,5> "\vec{a}-\vec{b}=<3-1,(-2)-(-7)> =<2,5>")
Dot Product
Given vectors
their dot product is defined as
which means that the dot product of two vectors produces a scalar.
Also, the following relationship is true for two vectors and the angle  between them:
Examples
1.
Find the dot product of the given vectors
Solution:
their product is
 (4)+ (5)(6)=-8+30=22 "\vec{a} \, \cdot \, \vec{b} =(-2) (4)+ (5)(6)=-8+30=22")
2. Find the angle between the vectors in the previous example
we know that
we also know from the previous example that
now,
^2+(5)^2}=\sqrt{29} "||\vec{a}|| = \sqrt{(-2)^2+(5)^2}=\sqrt{29}")
^2+(6)^2}=\sqrt{52} "||\vec{b}|| = \sqrt{(4)^2+(6)^2}=\sqrt{52}")
solving the dot product formula for we get
(\sqrt{52})}=59^o "\theta = \arccos{\frac{\vec{a} \, \cdot \, \vec{b}}{||\vec{a}||\,||\vec{b}||} = \arccos{\frac{22}{(\sqrt{29})(\sqrt{52})}=59^o")
hence the angle between the vectors is 59 degrees.
Finally, notice that if the angle between the vectors is 90 degrees, the value of the dot product will be zero since cos(90) = 0
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