aem ch9a stu

8/11/2019 AEM Ch9a Stu

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AEM 2, 9, JJEONGRF Design Lab. 1


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A scalar is a quantity that is determined by its magnitude.

Examples : time, temperature, length, distance, speed, density, energy, and voltage.

A vector is a quantity that has both magnitude and direction.

Examples : displacement, velocity, and force.We denote vectors by lowercase boldface letters a, b, v, etc.

In handwriting you may use arrows, for instance, (in place of a), , etc.

A vector (arrow) has a tail, called its initial point, and a tip, called its terminal point.

The length (or magnitude) of the vector a is denoted by |a|.

Another name for length is norm (or Euclidean norm).

A vector of length 1 is called a unit vector.


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AEM 2, 9, JJEONGRF Design Lab.

Two vectors a and b are equal, written a = b, if they have the same length and the same

direction.

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We choose anxyz Cartesian coordinate system in space. Let abe a given vector with

initial pointP: (x1,y1,z1) and terminal point Q: (x2,y2,z2). Then the three coordinatedifferences

a1 =x2 -x1, a2 = y2 - y1, a3 = z2 - z1

are called the components of the vector a with respect to that coordinate system, and we

write simply a = [a1, a2, a3].

The length |a| can be expressed in terms of components, we have

Position vector r of a pointA:(x, y, z) is the vector with the origin (0,0,0) as the initial point

andA as the terminal point. In components, r = [x, y, z]

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a3k


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Consider a force F acting on a rigid body at a point given by a position vector r.

Torque :

Tendency of the body to rotate about the origin

Cross product of the position and force vectors

Direction : the axis of rotation


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We define a vector function v, whose values are vectors, that is,

v = v(P) = [v1(P), v2(P), v3(P)] that depends on pointsP in space.

We say that a vector function defines a vector field.

Example of vector fields : gravitation field (Ex3).

Similarly, we define a scalar functionf, whose values are scalars, that is,f = f (P)

that depends onP. We say that a scalar function defines a scalar field.

Example of scalar fields : temperature field in space.

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