1. Multiplication with Vectors
Scalar Multiplication
Dot Product
Cross Product

2. Objectives
TSW use the dot product to fin the relationship between two vectors.
TSWBAT determine if two vectors are perpendicular

3. 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 _____________

4. Find the sum Vector a = < 3, 9 > and vector b = < -1, 6 >
What is the magnitude of the resultant.
Hint* remember use the distance formula.

5. Multiplication with Vectors
Scalar Multiplication
Dot Product
Cross Product

6. Scalar Multiplication:
returns a vector answer
Distributive Property:

7. Multiplication with Vectors
Scalar Multiplication
Dot Product
Cross Product

8. Dot Product Given and are two vectors,
The Dot Product ( inner product )of and is defined as
A scalar quantity

9. Finding the angle between two Vectors
a - b θ b

10. Example
Find the angle between the vectors:

11. 1:

13. 2:

16. 3:

17. Classify the angle between two vectors:
Acute : ______________________________________________
Obtuse: _____________________________________________
Right: (Perpendicular , Orthogonal) _______________________

18. example Given three vectors determine if any pair is perpendicular
THEOREM: Two vectors are perpendicular iff
their Dot (inner) product is zero.
Given three vectors determine if any pair is perpendicular

21. Ex 1:

23. Ex 2:

25. Ex 3:
Find the unit vector in the same direction as v = 2i-3j-6k

26. Ex 4: If v = 2i - 3j + 6k and w = 5i + 3j – k
Find:

27. Ex 5:
(c) 3v (d) 2v – 3w (e)

29. Ex 6: Find the angle between u = 2i -3j + 6k and v = 2i + 5j - k

32. Ex 7: Find the direction angles of
v = -3i + 2j - 6k

34. 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

35. 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.

36. Example Find the Dot Product of vector v and w.
Classify the angle between the vectors.

37. Projection of Vector a onto Vector b
Written :

38. Example:
Find the projection of vector a onto vector b :

39. 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

40. Multiplication with Vectors
Scalar Multiplication
Dot Product
Cross Product

41. OBJECTIVE 1

45. OBJECTIVE 2

47. OBJECTIVE 3

49. OBJECTIVE 4

53. OBJECTIVE 5

55. 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.

56. If
Then the cross product of vector a and vector b is defined as follows:

57. The determinant of a 2 x 2 matrix

58. 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.
You can check your answer by using the dot product.

59. Example Find the cross product of vector a and vector b if:
Verify that your answer is correct.

61. Assignment