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Classical Three-Space

Classical 3 Space

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Classical Three-Space (also known as Euclidean space) is a way of describing positions in space quantitatively. For classical mechanics, space is considered as being three-dimensional, so a position in space can be identified by three real numbers, usually called {$x, y, z$} or sometimes {$x_1, x_2, x_3 .$}

Classical 3 Space

The three numbers refer to distances along the axes of a Cartesian co-ordinate system, constructed with an arbitrary origin, or zero point, and three perpendicular axes.

Right-Hand Rule

Professor Susskind gives the pointy version of the right-hand rule. I prefer the curly finger version: if you curl the fingers of your right hand from the x-axis to the y-axis (that is, counter-clockwise, the same as the positive direction of angles), your thumb will be pointing in the direction of the z-axis. This is the same orientation as most standard screws: if you turn them counter-clockwise, they come out, which toward you.

Some 3D graphing calculators by default show the x-axis as coming toward the viewer, which means the z-axis is pointing up (and the y-axis points to the right). Professor Susskind draws 3D graphs with the z-axis coming out from the board, the y-axis up, and the x-axis to the right. These are completely compatible systems.

Beside an origin and the three axes, the co-ordinate system needs a unit of measure. For space the unit of measure is a measure of length such as meters, inches, miles, and so forth. Then every position in space can be represented by a distance on the x-axis, a distance on the y-axis, and a distance on the z-axis or {$(x, y, z) .$}

Vectors

A vector is a magnitude with an associated direction. Vectors in general do not have to be tied to the origin, but a position can be represented by a vector with a tail on the origin.

Vectors are represented on the blackboard with a bar or arrow over them {$\bar v, \overrightarrow v ,$} and in print by boldface {$\mathbf v.$} [More about vectors]

The length or magnitude of a vector is represented by enclosing it in bars or double bars: {$\rVert v \lVert, \rvert v \lvert.$} As might be expected of a length, this value is always a non-negative number (with length unit). The length is found by a three-dimensional extension of the Pythagorean theorem.

{$$\Large \lVert {\bf v} \rVert = \sqrt{v_x^2 + v_y^2 + v_z^2} \tag{length of a vector}$$}

The components {$v_x, v_y, v_x$} are not shown as vectors, because they are just numbers, lengths on their respective axes. Because they are just numbers, they can be squared in the ordinary way numbers are squared.

Operations on Vectors

A Vector can be multiplied by an ordinary number, called a scalar.

{$$\Large k {\bf v} = (kv_x, kv_y, kv_z) \tag{scalar multiplication}$$}

If a vector is multiplied by two, the result is a vector that is twice as long, and if multiplied by three, the result is three times as long. When a vector is multiplied by a negative number, the direction of the vector is changed to the opposite direction, but scalar multiplication cannot change the direction of a vector in any other way.

Two vectors can be added to produce another vector.

Geometrically, vectors can be added by placing their tails together and constructing a parallelogram, with the result being the diagonal of the parallelogram between them, or by placing the tail of one vector at the head of the other, with the result being the third side of the triangle thus formed from the tail of the first vector to the head of the second.

Algebraically,

{$$\Large \mathbf a + \mathbf b = ( a_x+b_x, a_y+b_y, a_z+b_z) \tag{vector sum}$$}

It should be clear from the algebra that it does not matter which vector is taken first: {$\mathbf a + \mathbf b = \mathbf b + \mathbf a ,$} and also that scalar multiplication is distributive over vector addition {$k (\mathbf a + \mathbf b) = k\mathbf a + k\mathbf b.$}

The dot product of two vectors is the sum of the arithmetic product of the corresponding coefficients on the basis vectors.

{$$\Large {\bf A} \cdot {\bf B} = A_x B_x + A_y B_y + A_z B_z \tag{Dot product}$$}

Through some algebra and the Law of Cosines, if {$\theta$} is the angle between the vectors, the dot product can be shown to be:

{$$\Large {\bf A} \cdot {\bf B} = \lVert {\bf A} \rVert \lVert {\bf B} \rVert \cos \theta \tag{Dot product}$$}

"Scalar Projection of A onto B"
Scalar projection of A on B

This amounts to the projection of {$\mathbf A$} onto {$\mathbf B$} multiplied by the length of {$\mathbf B ,$} and the form should make clear that it also can be the projection of {$\mathbf B$} onto {$\mathbf A$} multiplied by the length of {$\mathbf A .$}

An immediate and very useful result of this is that if the dot product is zero, the vectors are perpendicular (because the cosign of a right angle is zero).

The radius vector, traditionally represented as {$r$} is the vector from the origin to a particle in space. Almost always, physics is interested in motion, which is how particles move over time, so most of the time {$r$} is considered as a function of time, and its components are functions of time.

A common convention is that a dot over a function represents the first-order derivative of the function with respect to time and two dots represents the second-order derivative with respect to time.

{\Large \begin{align} \mathbf r (t) &= \left( r_x (t), r_y (t). r_z (t) \right) \tag{position}\\ \mathbf v (t) = \frac{d}{dt} \mathbf r (t) = \mathbf{\dot r} &= \left( \dot r_x, \dot r_y. \dot r_z \right) \tag{velocity} \\ \mathbf a (t) = \mathbf{\ddot r} &= \left( \ddot r_x, \ddot r_y. \ddot r_z \right) \tag{acceleration} \end{align}}

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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.