C-12-3The Dot Product - Calculus useful practice problems for exams. PDF

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Calculus useful practice problems for exams....

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C-12.3 The Dot Product

Instructor: Bhagi Anand

How do you find the dot product 𝒂 󰇍 ∙𝒃󰇍󰇍 of two vectors if you know their lengths and the angle between them? If 𝜃 is the angle between the vectors a and b, then 󰇍󰇍 ∙𝒃󰇍 = _____________________ (fill in the blank) 𝒂 How do you find the dot product 𝒂 󰇍 ∙𝒃󰇍󰇍 of two vectors if you know their components? If 󰇍𝒂 󰇍 =< 𝑎1 ,𝑎2 ,𝑎3 > and

𝒃󰇍 =< 𝑏1 ,𝑏2 ,𝑏3 > then

󰇍󰇍 ∙𝒃󰇍󰇍 = _____________________________ 𝒂

(fill in the blank)

Algebraic Definition of a Dot Product of two vectors:

Two dimension: The dot product is not a vector, but a scalar. For this reason it is called the scalar product (or inner product). For two dimensional vectors, the dot product is Example 1:

Geometric Definition of a Dot Product of two vectors:

1

Corollary: If 𝜃 is the angle between the vectors a and b, then

𝑐𝑜𝑠𝜃 = __________________

(fill in the blank)

2

Two non zero vectors a and b are called perpendicular or _________________ if the angle between them is _______________ Then theorem 3 gives a . b = _______________________= _____________

Two vectors 𝑎 and 𝑏󰇍 are orthogonal if and only if 󰇍𝒂 ∙𝒃󰇍 = 𝟎

(fill in the blank)

3

4

Vector projection of b onto a :

Vector projection of b onto a :

Scalar projection of b onto a :

Scalar projection of b onto a :

𝑐𝑜𝑚𝑝󰇍𝒂 󰇍𝒃 =

Vector projection of b onto a :

𝑝𝑟𝑜𝑗󰇍𝒂 󰇍𝒃 =

(fill in the blank) (fill in the blank)

Note:

5

If a force moves the object from P to Q, then the displacement vector is D = PQ . The work done by this force is defined to be the product of the component of the force along D and the distance moved:

6

How are dot products useful? • The dot product can be used to find the__________ between two vectors. (fill in the blank) • In particular, it can be used to determine whether two vectors are _______________ (fill in the blank) • The dot product can be used to find the ________________________of one vector onto another. (fill in the blank) • If a constant force moves an object, the _______________ is the dot product of the ____________and the________________________. (fill in the blank) Suggested problems from the exercise: *5)

*17)

*23) 7

*29)

*35)

*39) PT*41) PT*49)

*50)

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