Vectors. We will start with a basic review of vectors.
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Transcript of Vectors. We will start with a basic review of vectors.
Vectors
• We will start with a basic review of vectors.
• We will start with a basic review of vectors.• Recall: We can add vectors graphically.
a
b
a
ba+b
• However an easier way is to add components.• Recall that any vector can be written as x and
y components:jaiaa yxˆˆ
• However an easier way is to add components.• Recall that any vector can be written as x and
y components:
• wherejaiaa yxˆˆ
cosaax
sinaay
22yx aaa and
x
y
a
atan
Important concepts in vectors
Adding vectors by components
• Assume two vectors:jaiaa yxˆˆ
jbibb yxˆˆ
Adding vectors by components
• Assume two vectors:
• The sum of the two vectors is:
jaiaa yxˆˆ
jbibb yxˆˆ
jbaibaba yyxxˆˆ
Example
• Consider the vectors:
• Then,
jmima ˆ5.1ˆ2.4
jmimb ˆ9.2ˆ6.1
jmimba ˆ4.1ˆ6.2
Dot Product (Scalar product)
• The dot product between two vectors is defined as:
cos. abba
The smallest angle between the vectors
a
b
Dot Product (Scalar product)
• The dot product between two vectors is defined as:
• In unit vector notation:
cos. abba
The smallest angle between the vectors
kbjbibkajaiaba zyxzyxˆˆˆˆˆˆ.
zzyyxx bababa
a
b
Dot Product (Scalar product)
• The scalar produce of two vectors is a scalar!
Example
• Find the scalar product of the vectors, jmima ˆ5.1ˆ2.4 jmimb ˆ9.2ˆ6.1
Example
• Find the scalar product of the vectors, jmima ˆ5.1ˆ2.4 jmimb ˆ9.2ˆ6.1
jijiba ˆ9.2ˆ6.1ˆ5.1ˆ2.4.
jjijjiii ˆ9.2ˆ5.1ˆ6.1ˆ5.1ˆ9.2ˆ2.4ˆ6.1ˆ2.4 9.25.16.12.4
07.11
Vector Product (Cross product)
• The vector product between two vectors is defined as:
• The cross product of two vectors is a vector which is perpendicular to the plane of the vectors a and b.
sinabba
a
b
Vector Product (Cross product)
• The direction of the resultant vector is given by the right hand rule.
a
b
Using the right hand the fingers curl from the vector a to b while the thumb points in the direction of the resultant vector
ba
Vector Product (Cross product)
• The direction of the resultant vector is given by the right hand rule.
• In unit vector notation:
a
b
Using the right hand the fingers curl from the vector a to b while the thumb points in the direction of the resultant vector
ba
kabbajabbaiabbaba yxyxxzxzzyzyˆˆˆ