Cartesian Components - Self made notes from both the lectures and individual study. PDF

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Self made notes from both the lectures and individual study....

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Cartesian Components

Consider the x - y plane. The general point P has coordinates (x, y). P can be joined to the origin by a vector OP, which is called the position vector of P, which we denote by r. The modulus of r is |r| = r, and is the length of OP. It is possible to express r in terms of the numbers of x and y. If a unit vector is denoted along the x axis by i and a unit vector along the y axis by j (the ^ is usually omitted here), then from the definition of scalar multiplication that OM = xi, and MP = yj. It follows from the triangle law of addition that:

Examples: 󰇍󰇍󰇍󰇍󰇍 , where P is the point with coordinates (5, 1) and Q P and Q lie in the x - y plane. Find 𝑃𝑄 󰇍󰇍󰇍󰇍󰇍 | is the point with coordinates (-1, 4). Find |𝑃𝑄 Solution:

󰇍󰇍󰇍 󰇍󰇍 + 𝑃𝑄 󰇍󰇍󰇍󰇍󰇍 = 󰇍󰇍󰇍 󰇍󰇍󰇍 𝑂𝑄 𝑂𝑃 󰇍󰇍󰇍 󰇍󰇍 = 󰇍󰇍󰇍󰇍 󰇍󰇍 − 𝑂𝑃 󰇍󰇍󰇍󰇍󰇍 𝑂𝑄 𝑃𝑄

−6 −1) − (5 )=( ) 3 4 1 󰇍󰇍󰇍󰇍󰇍 | = √(−6)2 + (3)2 = √45 |𝑃𝑄 (

Always follow the triangle rule. The vectors i and j are orthogonal. The numbers 𝑥 and y are the i and j components of r. Using Pythagoras's theorem:

Alternative notations:

This is known as a column vector. This is known as a row vector.

Cartesian Components

To find AB, use Pythagoras's Theorem, that is, √𝑎2 + 𝑏2 = √68 = 8.25

Cartesian Components

Cartesian Components Linear combinations, dependence and independence If vectors a and b have arbitrary scalar multiples, that is 𝑘1 𝑎 and 𝑘2 𝑏, and these are added together, a new vector c is formed where c = 𝑘1 𝑎 + 𝑘2 𝑏. The vector c is said to be a linear combination of a and b. Scalar multiplication and addition of vectors are the only operations allowed when forming a linear combination. Vector c is said to depend linearly on a and b. This could also be written in this form, provided 𝑘2 ≠ 0: 𝑏=

𝑘1 1 𝑐− 𝑎 𝑘2 𝑘2

So that b depends linearly on c and a. The set of vectors {a, b, c} is said to be linearly dependent and any one of the set can be written as a linear combination of the other two.

Cartesian Components Planes A plane is two dimensional and so it will have two direction vectors. The Zero Vector A vector where all the components are zero and is denoted by 0 to distinguish it from the scalar 0. The zero vector has a length of zero; it is unusual in that it has arbitrary direction. Lines A line can be described by two vectors. One gives a point on the line, the other gives a direction parallel to the line. The general form of a vector equation of a line is: 𝑥 = 𝑥0 + 𝑡𝑣 Where x represents an arbitrary point on the line, 𝑥0 , is a fixed point on the line, v is a vector parallel to the line, and t is a parameter.

A line can be represented as 𝑦 = 𝑚𝑥 + 𝑐 and this is similar to the general equation of a line: 𝐴𝑥 + 𝐵𝑦 = 𝐶

Cartesian Components The Unit Vector Vectors which have length 1 are called unit vectors. If a has length 3 for example, then a unit vector 1 in the direction of a is 𝑎. 3

 𝒂=

𝒂 |𝒂|

3

𝑣 1 3 For example: let v be ( ). Then |𝑣| = √32 + 42 = √25 = 5. So 𝑣 = = 5 𝑣 = ( 54) 5 4 5

Orthogonal vectors If the angle between two vectors a and b is 90°, that is a and b are perpendicular, then a and b are said to be orthogonal. To test if a and b are orthogonal, take the dot product. If 𝑎 ⋅ 𝑏 = 0, then a and b are orthogonal.

Cartesian Components

Cartesian Components

The Vector/Cross Product The result of finding a vector product of a and b is a vector of length |𝑎||𝑏| sin 𝜃 , where θ is the angle between a and b. The direction of this vector is perpendicular to a and b, therefore it is perpendicular to the plane containing a and b. Two possible directions exist for this vector, but it is better to associate it with the application of the right - handed screw rule. Imagine turning a right - handed screw in the sense from a towards b as shown. The direction in which the screw advances is the direction of the required vector product. The symbol we shall use to denote the vector product is ×. Formally, we write

𝑎 × 𝑏 = |𝑎||𝑏| sin 𝜃𝑒

Cartesian Components

Cartesian Components

Cartesian Components Scalar fields and Vector fields Imagine a large room filled with air. At any point, P, we can measure the temperature, φ, say. The temperature will depend upon whereabouts in the room we take the measurement. Perhaps, close to a radiator the temperature will be higher than near to an open window. Clearly the temperature φ is a function of the position of the point. If we label the point by its Cartesian coordinates (x, y, z), then φ will be a function of x, y and z, that is φ = φ(x, y, z) Additionally, φ may be a function of time but for now we will leave this additional complication aside. Since temperature is a scalar what we have done is dene a scalar at each point P(x, y, z) in a region. This is an example of a scalar field. Alternatively, suppose we consider the motion of a large body of fluid. At each point, fluid will be moving with a certain speed in a certain direction; that is, each small fluid element has a particular velocity, v, depending upon whereabouts in the fluid it is. Since velocity is a vector, what we have done is dene a vector at each point P(x, y, z). We now have a vector function of x, y and z, known as a vector field. Let us write 𝑣 = (𝑣𝑥 , 𝑣𝑦 , 𝑣𝑧 )

So that 𝑣𝑥 , 𝑣𝑦 𝑎𝑛𝑑 𝑣𝑧 are the i, j and k components respectively of v, that is 𝑣 = 𝑣𝑥 𝑖 + 𝑣𝑦 𝑗 + 𝑣𝑧 𝑘

Note that 𝑣𝑥 , 𝑣𝑦 𝑎𝑛𝑑 𝑣𝑧 will each be scalar functions of 𝑥, 𝑦 𝑎𝑛𝑑 𝑧. The scalar product Given any two vectors a and b, two ways exists in which we can define their product: the Scalar product and the Vector product. As names suggests, the result of finding a scalar product is a scalar whereas the result of finding a vector product is a vector. The scalar product of a and b is written as a.b This notation gives rise to the alternative name dot product. It is defined by the formula 𝑎 ⋅ 𝑏 = |𝑎||𝑏| 𝑐𝑜𝑠 𝜃

Where 𝜃 is the angle between the two vectors as shown.

Cartesian Components The following rules hold for the scalar product:

Cartesian Components

Using determinants to evaluate vector products A convenient and easy method, the vectors a and b are written in the following pattern:

𝒊 𝑎1 𝑏1

𝒋 𝑎2 𝑏2

𝒌 𝑎3 𝑏3

To find component i, imagine crossing out the row and column containing i and performing the following calculation on what is left, that is: 𝑎2 𝑏3 − 𝑎3 𝑏2 The resulting number is the i component of the vector product. The j component is found by crossing out the row and column containing j, performing a similar calculation, but now changing the sign of the result. Thus the j component equals:

−(𝑎1 𝑏3 − 𝑎3 𝑏1 )

The k component is found out through crossing out the row and column containing k and performing the calculation:

𝑎1 𝑏2 − 𝑎2 𝑏1...