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