Upload
carver
View
38
Download
0
Tags:
Embed Size (px)
DESCRIPTION
Multiplication with Vectors. Scalar Multiplication Dot Product Cross Product. Objectives. TSW use the dot product to fin the relationship between two vectors. TSWBAT determine if two vectors are perpendicular. A bit of review. A vector is a _________________ - PowerPoint PPT Presentation
Citation preview
Multiplication with Vectors
Scalar MultiplicationDot Product
Cross Product
Objectives
• TSW use the dot product to fin the relationship between two vectors.
• TSWBAT determine if two vectors are perpendicular
A bit of review
• A vector is a _________________
• The sum of two or more vectors is called the ___________________
• The length of a vector is the _____________
Find the sum
• Vector a = < 3, 9 > and vector b = < -1, 6 >
• Find
• What is the magnitude of the resultant.
• Hint* remember use the distance formula.
a b
Multiplication with Vectors
Scalar MultiplicationDot Product
Cross Product
Scalar Multiplication: returns a vector answer
Distributive Property:
If 4 6 find 2a i j a
If 3 2 b 5 find 2 3a i j i j a b
Multiplication with Vectors
Scalar MultiplicationDot ProductCross Product
Dot Product
• Given and are two vectors, • The Dot Product ( inner product )of and is defined
as
• A scalar quantity
1 2( , )a a a1 2( , )b b b
a b
1 1 2 2a b a b a b
Finding the angle between two Vectors
a
b
θa - b
2 22
22
22
2 22 2
2 cos( )
2 cos( )
2 2 cos( )
2 2 cos( )
cos( )
a b a b a b
a b a b a b a b
a a a b b b a b a b
a a b b a b a b
a ba b
Example• Find the angle between the vectors:
3,6 and 4, 2r s
1:
2:
3:
Angle between Vectors
cos( ) a ba b
Classify the angle between two vectors:
Acute : ______________________________________________
Obtuse: _____________________________________________
Right: (Perpendicular , Orthogonal) _______________________
example
• Given three vectors determine if any pair is perpendicular
3,12a 8, 2b 3,2 c
THEOREM: Two vectors are perpendicular iff their Dot (inner) product is zero.
Ex 1:
Ex 2:
Find the unit vector in the same direction as v = 2i-3j-6k
Ex 3:
Ex 4: If v = 2i - 3j + 6k and w = 5i + 3j – k
Find: ( ) (b) w v (c) v v
(d) w w (e) v (f) w
a v w
(c) 3v (d) 2v – 3w (e) v
Ex 5:
Ex 6: Find the angle between u = 2i -3j + 6k and v = 2i + 5j - k
Ex 7: Find the direction angles of
v = -3i + 2j - 6k
Ex 8: The vector v makes an angle of with the 3
positivex-axis, an angle of = with the positive y-axis,3
and an acute angle with the positive z-axis. Find .
Any nonzero vector v in space can be written in terms of its magnitude and direction cosines as:
Ex 9: Find the direction angles of the vector below. Write the answer in the form of an equation.
v = 3i – 5j + 2k 9 25 4 38V
5cos ; 14438
3cos ; 6138
2cos ; 7138
38 cos 61 cos144 cos 71i j k
• We can also find the Dot Product of two vectors in 3-d space.
• Two vectors in space are perpendicular iff their inner product is zero.
1 1 2 2 3 3a b a b a b a b
Example
• Find the Dot Product of vector v and w.
• Classify the angle between the vectors.
6,2,10 and 4,1,3v w
Projection of Vector a onto Vector b
a
ba
b
Written : bproj a
2 ba bproj a bb
Example:
Find the projection of vector a onto vector b :
3 and - 2 4a i j b i j
4,1, 1 and 2, 3, 3a b
Decompose a vector into orthogonal components…
Find the projection of a onto b
Subtract the projection from a
The projection, and a - b are orthogonal
a
b
a-b bproj a
bproj a
Multiplication with Vectors
Scalar MultiplicationDot Product
Cross Product
OBJECTIVE 1
find v w
OBJECTIVE 2
OBJECTIVE 3
OBJECTIVE 4
OBJECTIVE 5
Cross product• Another important product for vectors in
space is the cross product. • The cross product of two vectors is a vector.
This vector does not lie in the plane of the given vectors, but is perpendicular to each of them.
• If
1 2 3
1 2 3
, ,
, ,
a a a a
b b b b
Then the cross product of vector a and vector b is defined as follows:
2 3 1 3 1 2
2 3 1 3 1 2
a a a a a aa b i j k
b b b b b b
The determinant of a 2 x 2 matrix
1 11 2 1 2
2 2
= - a b
a b b aa b
2 5 =
3 4
• An easy way to remember the coefficients of vectors I, j, and k is to set up a determinant as shown and expand by minors using the first row.
1 2 3
1 2 3
i j ka a ab b b
You can check your answer by using the dot product.
2 3 1 3 1 2
2 3 1 3 1 2
a a a a a ai j k
b b b b b b
Example• Find the cross product of vector a and vector b
if: 5,2,3 2,5,0a b
Verify that your answer is correct.
Assignment