College Pre-Calc
Section 12.4: Parallel and Perpendicular Vectors:
Dot Product
Definition of the Dot Product:
a • b = ( a1 , a2 ) • ( b1 , b2 ) = a1b1 + a2b2
➔ also known as the scalar product or inner product
➔ a • b is a one "number" answer
College Textbook - "Chapter 7"
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College Pre-Calc
Section 12.4: Parallel and Perpendicular Vectors: Dot Product
Properties of the Dot Product: If a , b , and
c are vectors and m is a real number then:
1) a • a = | a |2
2) a • b = b • a
3) a • ( b + c ) = a • b + a • c
4) m(a • b ) = (ma) • b = a • (mb)
College Textbook - "Chapter 7"
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College Pre-Calc
Section 12.4: Parallel and Perpendicular Vectors: Dot Product
The Angle Between Two Vectors:
If θ is the angle between two nonzero vectors a
and b, then:
cos θ =
a •b
|a |*|b |
in other words...
cos θ =
College Textbook - "Chapter 7"
a1b1 + a2b2
√a12+a22 *√b12+b22
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College Pre-Calc
Section 12.4: Parallel and Perpendicular Vectors: Dot Product
Orthogonal Vectors:
Two vectors are orthogonal (perpendicular) if and
only if a • b = 0
in other words...
two vectors are perpendicular if their
DOT PRODUCT is ZERO
Parallel Vectors:
Two vectors are parallel if they are "multiples" of
each other
College Textbook - "Chapter 7"
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College Pre-Calc
Section 12.4: Parallel and Perpendicular Vectors: Dot Product
Example: Let a = ( 8 , -4 ) and b = ( 2 , 1 ).
a) Show that a • b = b • a
b) Find the angle θ between a and b to the nearest
tenth of a degree.
c) Find a vector that is parallel to a.
d) Find a vector that is perpendicular to a.
College Textbook - "Chapter 7"
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College Pre-Calc
College Textbook - "Chapter 7"
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