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Vectors and Vector MultiplicationPowerPoint Presentation

Vectors and Vector Multiplication

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Vector quantities are those that have magnitude and direction, such as:

- Displacement, x or
- Velocity,
- Acceleration,
- Force,
- Torque,
- Electric field, ….to name just a few

Scalar quantities have only magnitude: direction, such as:

- Speed, v
- Distance, d
- Time, t
- Energy, E
- Power, P
- Charge, q
- Electric potential, V

Multiplication of scalar quantities follows all the “usual” rules, including:Distributive a(b+c) = ab + acCommutative ab = baAssociative (ab)c = a(bc)

Addition of scalars follows these properties: “usual” rules, including:Commutative a+b = b+aAssociative (a+b)+c = a+(b+c)Subtraction a+(-b) = a-b

Addition of vectors is commutative and associative and follows the subtraction rule:A+B = B+A(A+B)+C = A+(B+C)A-B = A+(-B)

Multiplication of a scalar and a vector follows previous rules:aB = Baa(B+C) = aB + aC

However rules:, multiplication of vectors has a new set of rules—the vector cross product (or “vector product”) and the vector dot product or “scalar product”.

Vector Dot Product rules:or Scalar ProductA·B = AB cosEssentially, this means multiplying the first vector times the component of the second vector that is in the same direction as the first vector—yielding a product that is a scalar quantity.

B rules:

B sin

A

B cos

A·B = AB cos

Multiple the magnitude of vector A times the magnitude of vector B times the cosine of the angle between them—or multiply the components that are in the same direction. The answer is a scalar with the units appropriate to the product AB.

Vector Cross Product rules:or Vector ProductAxB = AB sinEssentially, this means multiplying the first vector times the component of the second vector that is perpendicular to the first vector—yielding a product that is a vector quantity. The direction of the new vector is found using the right hand rule.

B rules:

B sin

A

B cos

Multiple the magnitude of vector A times the magnitude of vector B times the sine of the angle between them—or multiply the components that are perpendicular. The answer is a vector with the units appropriate to the product AB and direction found by using the right hand rule.

For example, let’s take the vector cross product: rules:

F = q (vxB)

where q is the charge on a proton, v is 3x105 m/s to the left on the paper, and B is 500 N/C outward from the paper toward you. The equation for this is also: F = qvB sin

The answer for the force is 2.4 x 10 rules:-11 newtons toward the top of the paper.

Unit vectors rules:

Unit vectors have a size of “1” but also have a direction that gives meaning to a vector.

We use the “hat” symbol above a unit vector to indicate that it is a unit vector.

For example, is a vector that is 1 unit in the x-direction. The quantity 6 meters is a vector 6 meters long in the x-direction.

Did you realize that you have been using a right-handed Cartesian coordinate system in mathematics all these years?

You can check your use of the right hand rule, because Cartesian coordinate system in mathematics all these years?

Here are a few for practice: Cartesian coordinate system in mathematics all these years?

We can also do Cartesian coordinate system in mathematics all these years?dot products with unit vectors. Try these:

1

0

0

1

8

12 m2

The calculation of work is a scalar product or dot product: Cartesian coordinate system in mathematics all these years?

What is the work done by a force of 6 newtons east on an object that is displaced 2 meters east?

What is the work done by a force of 6 newtons east on an object that is displaced 2 meters north?

What is the work done by a force of 6 newtons east on an object that is displaced 2 meters at 30 degrees north of east?

12 joules

zero

10.4 joules

In summary: Cartesian coordinate system in mathematics all these years?

- In an equation or operation with a scalar or dot product, the answer is a scalar quantity that is the product of two vectors.
- The dot product is found by multiplying the components of vectors that are in the same direction.
- In an equation or operation with a vector or cross product, the answer is a vector quantity that is the product of two vectors.
- The cross product is found by multiplying the components of vectors that are perpendicular to each other.

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