Vector and Scalar Product

Polygon Law of vector addition

The sum of more than two vectors is obtained in accordance with the polygon law of addition which states that if a vector polygon be drawn, placing the tail of each succeeding vector at the head of preceding one. Then their resultant \(\vec R\) is drawn from the tail of the first vector to the head of the last vector as shown in the figure

$$ \vec R = \vec A + (\vec B + \vec C ) = (\vec A + \vec B)+ \vec C $$

This shows that vector addition is associative

Laws of Vector Algebra

If \(\vec A,\: \vec B,\:\vec C\) are vectors and k, s are scalars then

1. Commutative law of Addition
   $$\vec A + \vec B = \vec B + \vec A $$
2. Associative law of Addition
   $$ \vec A + (\vec B + \vec C ) = (\vec A + \vec B)+ \vec C $$
3. Commutative law of multiplication
   $$\vec k A = \vec A m $$
4. Associative law of Multiplication
   $$k(s \vec A) = (ks)\vec A$$
5. Distributive law of Addition
   $$ (k + s) \vec A = k\vec A + s\vec A $$
6. Distributive law of multiplication

$$ s(\vec A + \vec B) = s\vec A + A\vec b$$

Orthogonal resolution of a vector

Let OA represents a given vector \(\vec a\) defined by three mutually perpendicular axes x, y, z.

$$\text {Then},\: \vec a = x\hat I + y\hat j + z\hat k $$

Where \(\hat I,\: \hat j\) and \(\hat k\) are unit vectors along x, y and x axis.

\begin{align*} |\vec a|&= \sqrt {x^2 + y^2 + z^2} \\ \text {Now}\: \hat a &= \frac { x\hat I + y\hat j + z\hat k }{\sqrt {x^2 + y^2 + z^2} }\\ \hat a &= \frac {x}{\sqrt {x^2 + y^2 + z^2} }\hat I +\frac {y}{\sqrt {x^2 + y^2 + z^2} }\hat j + \frac {z}{\sqrt {x^2 + y^2 + z^2} }\hat k \\ &= \cos \alpha \hat I +\cos \beta \hat j+\cos \gamma \hat k \\ \end{align*}

Where \(\cos \alpha, \: \cos \beta ,\: \cos \: \gamma \) are direction cosines of OA with x, y and z axes respectively. But

\begin{align*} |\hat a| = \sqrt {\cos ^2\alpha + \cos ^2\beta + \cos ^2 \gamma } &= 1 \\ \text {or,}\: \sqrt {\cos ^2\alpha + \cos ^2\beta + \cos ^2 \gamma} &= 1\end{align*}

Thus sum of square of three direction cosines of a vector is equal to unity.

1. Product of two vectors

   Dot product or scalar product: The dot or scalar Product of two vectors \(\vec A\) and \(\vec B\), denoted by \(\vec A . \vec B\) is defined as the product of the magnitudes of \(\vec A\) and \(\vec B\) and the cosines of the angle \(\theta \) between them.
   The following laws are valid:
   1. \(\vec A.\vec B =\vec B.\vec A\) commutative law
   2.\( \vec A. (\vec B + \vec C) = \vec A. \vec B + \vec A. \vec C \) distributive law
   3. \(m(\vec A.\vec B) =(m\vec A).\vec B = \vec A. (m\vec B) = (\vec A.\vec B)M,\) where m is a scalar.
   4. If \(\vec A = A_1\vec i + A_2\vec j + A_3\vec k \) and \(\vec B = B_1\vec I + B_2\vec j + B_3\vec k,\) then
   $$\vec A.\vec B =A_1.B_1 + A_2B_2 + A_3B_3 $$
   $$\vec A.\vec A = A^2 = A_1^2 + A_2^2 + A_3^2$$
   $$\vec B. \vec B = B^2 =B_1^2 + B_2^2 + B_3^2$$
   6. If \( \vec A.\vec B = 0 \) and \(\vec A \) and \(\vec B \) are null vetors, then \(\vec A \) and \(\vec B\) are perpendicular.

1. Cross or vector product:

   The cross or vector product of \(\vec A\) and \(\vec B\) is a vector \(\vec C = \vec A \times \vec B\). The magnitude of \(\vec A \times \vec B \) is defined as the product of magnitude of \(\vec A\) and \(\vec B\) and the sine of the angle \(\theta \) between them. The direction of the vector \(\vec C = \vec A \times \vec B\) is perpendicular to plane of \(\vec A \) and \(\vec B\) such that \(\vec A \times \vec B\) and \(-\vec B \times\vec A \) form a right handed system.
   1. \(\vec A\times \vec B =-\vec B \times\vec A\) commutative law
   2.\( \vec A\times (\vec B + \vec C) = \vec A\times \vec B + \vec A\times \vec C \) distributive law
   3. \(m(\vec A \times\vec B) =(m\vec A)\times\vec B = \vec A \times(m\vec B) = (\vec A\times \vec B)M,\) where m is a scalar.
   4.\(\vec i\times \vec I = \vec j\times \vec j = \vec k\times \vec k = 0,\:\:\: \vec i\times \vec j = \ vec j\times \vec k = \vec k \times \vec i = 0\)
   5. If \(\vec A = A_1\vec i + A_2\vec j + A_3\vec k \) and \(\vec B = B_1\vec I + B_2\vec j + B_3\vec k,\) then
   \begin{matrix} \vec A \times \vec B = &\hat i &\hat j &\hat k \\ &A_1 &A_2 &A_3 \\ &B_1 &B_2 &B_3 \end{matrix}
   6. If \( \vec A\times \vec B = 0 \) and \(\vec A \) and \(\vec B \) are not null vectors, then \(\vec A \) and \(\vec B\) are perpendicular.

Triple Product

1. Scalar triple product

   Let the vectors \(\vec A, \: \vec B\: \text {and}\: \vec C\) represent the edges of a parallelepiped is a vector normal to the plane of \(\vec B\) and \(\vec C\). But the area of base of the parallelepiped \(=BC\: \sin \phi \) which is the magnitude of vector product \(\vec B\times \vec C\) and its direction is perpendicular to this area.
   The scalar product of \(\vec A\) with \((\vec B \times \vec C )\) is the product of this area and the projection of \(\vec A\) along \((\vec B \times \vec C )\) . The projection of \(\vec A\) along \((\vec B \times \vec C )\: A\cos \theta\), i.e. the vertical height of parallelepiped.
   
   Geometrical representation of scalar triple product:
   Scalar triple product represents volume of a parallelepiped. Let OA, OB and OC be three coterminous sides of a parallelepiped as shown in figure.

   

   
   \begin{align*} \vec {OA} = \vec a,\: \vec {OB} = \vec b\: \text {and}\: \text {OC} = \vec c \\ \text {The area of parallelogram OBDC} =|\vec b \times \vec a |\\ \end{align*}
   Let \(\widehat n\) be unit vector perpendicular to the plane of \(\vec b\) and \(\vec c\) such that \(\widehat n\) , \(\vec b\) and \(\vec c\) for right handed system. Let h be the height of the terminal point of \(\vec a\) above the parallelogram OBDC. If \(\phi \) be the angle between \(\vec a\) and \(\widehat n\) then
   \begin{align*} h &= |\vec a|\cos \phi \\ &= |\vec a\cos \phi |\widehat n| \\ &= |\vec a|\:|\widehat n|\: \cos \phi = \vec a.\widehat n \\ \text {Now, volume of the parallelogram} (V) &= h \times \text {area of parallelogram OBCD} \\ &= h(|\vec b \times \vec c|) \\ V &= (\vec a.\widehat n) (|\vec b \times \vec c|) =\vec a. (|\vec b \times \vec c|\widehat n) \\ V &= \vec a. (\vec b \times \vec c) \end{align*}

2. Vector triple product

   The cross product \((\vec B \times \vec C )\) is a vector normal to the plane containing \(\vec B\) and \(\vec C\). Therefore \(\vec B \times (\vec B \times \vec C )\) will be perpendicular to the plane containing \(\vec A\) and \((\vec B \times \vec C )\). It will lie in the plane of \(\vec B\) and \(\vec C\) as shown in figure.Vector triple product
   The cross product \((\vec B \times \vec C )\) is a vector normal to the plane containing \(\vec B\) and \(\vec \). Therefore \(\vec B \times (\vec B \times \vec C )\) will be perpendicular to the plane containing \(\vec A\) and \((\vec B \times \vec C )\) e. it will lie in the plane of \(\vec B\) and \(\vec C\) as shown in figure.

   

Ordinary Differentiation of Vectors

Differentiation formula: If \(\vec A, \: \vec B \text {and}\:\vec C\) are differentiable vector function od a scalar u, and \(\phi \) is a differential scalar function of u, then the derivative of \(\vec R (u)\) with respect to u is defined as

\begin{align*}\frac {d\vec R}{du} &= \lim_{\Delta\: u\to{0}}\frac {\Delta \vec {R}}{\Delta u} \\ &= \lim_{\Delta\: u\to{0}} \frac {\vec R (u + \Delta u)- \Vec R (u)}{\Delta u}\\ \text {if limit exits}\end{align*}

Since \(\frac {d\vec R}{du}\) is itself a vector depending on u, we can consider its derivative with respet to u, if limit exits whih is denoted by \(\frac {d^\vec R}{du^2} \).

Differentiation formula

If \(\vec A,\: \vec B\: \text {and}\: \vec C \) are differentiable vector functions of a scalar u, and \(\phi \) is a differentiable scalar function of u, then

\(\frac {dC}{du} = 0\)

\(\frac {d}{du} (\vec A + \vec B) = \frac {d\vec A}{du} + \frac {d\vec B}{du}\)

\(\frac {d}{du} (\vec \vec B) = \vec A. \frac {d\vec B}{du} + \frac {\vec A}{du}. \vec B\)

\(\frac {d}{du} (\vec A \times \vec B) = \vec A. \frac {d\vec B}{du} + \frac {\vec A}{du}\times \vec B\)

\(\frac {d}{du}(\phi \vec A) = \phi \frac {d\vec A}{du} + \frac {d\phi }{du}\vec A\)

\(\frac {d}{du}(\vec A. \vec B \times \vec C) = \vec A. \vec B \frac {d\vec C}{du}+ \vec A. \frac {d\vec B}{du} \times \vec C + \frac {d\vec A}{du}. \vec B \times \vec C \)

\(\frac {d}{du} [\vec A \times (\vec B \times \vec C)] = \vec A \times (\vec B \times \frac {d\vec C}{du}) + \vec A \times (\frac {d\vec B }{du}\times \vec C) + \frac {d\vec A}{du}\times (\vec B \times \vec C)\)

Partial Derivative of Vectors

Scalar and Vector Point Functions

1. Field:
   If a function is defined in any region of space, for every point of the region then this region is known as the field.
2. Scalar point function:
   A function \(\phi (x, y, z\) is called scalar point function if it associates with every point in space. The temperature distribution in a heated body, density of a body, potential due to gravity et are examples of scalar point function.
3. Vector point function:
   A function \(\phi (x, y, z\) defines a vector at every point of a region then A function \(\phi (x, y, z\) is called a vector point function. Example the velocity of a moving fluid, gravitational force etc.
4. Vector differential operation (del):
   It is written as \(\del \), is defined in artesian co-ordinate system by
   \begin{align*}\del &= \frac {\partial}{\partial x}\hat i + \frac {\partial }{\partial y}\hat j + \frac {\partial}{\partial z} \hat k \\ &= \hat I \frac {\partial }{\partial x} + \hat j \frac {\partial }{\partial y} + \hat k \frac {\partial }{\partial z} \end{align*}
   The vector operator defines three quantities which arise in practical applications and are known as the gradient, the divergence and the curl.

Bibliography

P.B. Adhikari, Bhoj Raj Gautam, Lekha Nath Adhikari. A Textbook of Physics. kathmandu: Sukunda Pustak Bhawan, 2011.

Jha, V. K.; 'Lecture title'; Elementary Vector Analysis; St. Xavier's College, Kathmandu; 2016.