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- StudyBlue
- North-carolina
- University of North Carolina - Charlotte
- Mathematics
- Mathematics 2241
- Gordon
- 10.3 - The Dot Product

Jason S.

If **a** =〈a_{1}, a_{2}, a_{3}〉and **b** =〈b_{1}, b_{2}, b_{3}〉, then the **dot product** of **a** & **b** is the number **a•b** given by **a•b:**

- a
_{1}b_{1}+ a_{2}b_{2}+ a_{3}b_{3} - a
_{1}b_{1}− a_{2}b_{2}− a_{3}b_{3}

a_{1}b_{1} **+** a_{2}b_{2} **+** a_{3}b_{3}

The dot product is sometimes called the _______ product

- vector
- scalar

scalar product

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)

〈a_{1}, a_{2}〉•〈b_{1}, b_{2}〉=

- a
_{1}b_{1}+ a_{2}b_{2} - a
_{1}b_{1}− a_{2}b_{2}

a_{1}b_{1} + a_{2}b_{2}

- 2
**a** - |
**a**|^{2}

|**a**|^{2}

**b**•**a**- |
**a**|^{2}

- (
**a**+**b**)**c**

**a**•**b**+**a**•**c**

*c*(**a • b**) =**a •**(*c***b**)*c***ab**

**a****0**

0

Which is the Law of Cosines?

- c
^{2}= a^{2}+ b^{2} - a
^{2}= b^{2}+ c^{2}sinθ - a
^{2}= b^{2}+ c^{2}− 2bccosθ

a^{2} = b^{2} + c^{2} − 2bccos*θ*

Where a is the side of the triangle that *θ* is pointing at.

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

**a • b**= |**a**| |**b**| cos*θ***a • b**= |**a**| |**b**| sin*θ*

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

- cos
*θ*= (**a • b**) / (**a • b**) - cos
*θ*= (**a****•****b**) / (|**a**|**•**|**b**|)

cos *θ* = (**a • b**) / (|**a**| **•** |**b**|)

Two vectors **a** and **b** are _________ if and only if **a • b** = **0**

- parallel
- orthogonal

orthogonal

The dot product **a • b** is ________ if **a** and **b** point in the same general direction

- positive
- negative
- zero

positive

The dot product **a • b** is ________ if **a** and **b** are perpendicular (orthogonal).

- positive
- negative
- zero

zero

The dot product **a • b** is ________ if **a** and **b** point in general opposite directions.

- positive
- negative
- zero

negative

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In the extreme case where **a** and **b** point in exactly the same direction, we have *θ* = 0, so cos *θ* = 1 and

**a**•**b**= |**a**| |**b**| sin*θ***a**•**b**= |**a**| |**b**|

If **a** and **b** point in exactly opposite directions, then *θ *= π and so cos *θ* = −1 and**a **•** b** = |**a**| |**b**|**a **•** b** = −|**a**| |**b**|

Two nonzero vectors **a** and **b** are called ___________ or **orthogonal** if the angle between them is *θ* = π/2.

- perpendicular
- parallel

perpendicular

Two nonzero vectors a and b are called perpendicular or orthogonal if the angle between them is θ = π/2. Then Theorem 3 gives:

**a**•**b**= |**a**| |**b**| cos (π/2) = 0**a**•**b**= |**a**| |**b**| cos (0) =**a**•**b**= |**a**| |**b**|

If **a** • **b** > 0*θ* = acute*θ* = obtuse*θ* = π/2

If **a** • **b** = 0*θ* = acute*θ* = obtuse*θ* = π/2

If **a** • **b** < 0*θ* = acute*θ* = obtuse*θ* = π/2

proj_{a} **b**

(You can think of it as a shadow of **b**).

Scalar projection of b onto a:

- comp
_{a}**b**= (**a • b**) / |**a|** - proj
_{a}**b**= ((**a • b**) / |**a**|) (**a**/ |**a**| ) = ( (**a • b**) / |**a**|^{2})**a**

comp_{a}**b** = (**a • b**) / |**a**|

Vector projection of b onto a:

- comp
_{a}**b**= (**a • b**) / |**a**| - proj
_{a}**b**= ((**a • b**) / |**a**|) (**a**/ |**a**| ) = ( (**a • b**) / |**a**|^{2})**a**

proj_{a}**b** = ((**a • b**) / |**a**|) ( **a** / |**a**| ) = ( (**a • b**) / |**a**|^{2} )**a**

Notice that the vector projection is the ______ projection times the unit vector in the direction of **a**.

- vector
- scalar

scalar

**Scalar proj.** = comp_{a}**b** = (**a • b**) / |**a**|**Unit vector** = **a** / |**a**|

Vector = scalar × unit vector

= ((**a • b**) / |**a**|) ( **a** / |**a**| ) = ( (**a • b**) / |**a**|^{2} )**a**

Work = Force × Distance

From Theorem 3, we have:

**W**= |**F**| |**D**| sin*θ*=**F • D****W**= |**F**| |**D**| cos*θ*=**F • D**

Thus the work done by a constant force **F** is the dot product **F•D**, where **D** is the displacement vector.

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