Post on 22-Feb-2016
description
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