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General Physics I:

Subsection 2.1.2 Product of Two Vectors:

There are two ways in which vectors can be multiplied. The product of one of them is a scalar quantity, called a scalar product (or a dot product) and the other is a vector quantity, called a vector product (or a cross product).

Subsubsection 2.1.2.1 Scalar (or Dot) Product

The scalar product of two vectors \(\vec{a}\) and \(\vec{b}\) is denoted by \(\vec{a}\cdot \vec{b}\text{,}\) and is defined as
\begin{equation*} \vec{a}\cdot {\vec{b}} = |\vec{a}| |\vec{b}| \cos\theta = ab\cos\theta, \end{equation*}
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\text{.}\)

Geometrical Interpretation:.

Let \(\vec{OA} =\vec{a}\) and \(\vec{OB} =\vec{b}\text{,}\) as shown in Figure 2.1.3, then
\begin{equation*} \vec{a}\cdot {\vec{b}} = \vec{OA}\cdot {\vec{OB}} = (OA)(OB) \cos\theta = (OA)(ON) \end{equation*}
= (length of \(\vec{a}\)) (projection of \(\vec{b}\) along \(\vec{a}\)).
Figure 2.1.3.
Therefore the dot product of two vectors is the product of length of one of these vectors and the projection of the other in the direction of the first.
Mnemonics:
\begin{equation*} \hat{i}\cdot\hat{j}= \hat{j}\cdot\hat{k} = \hat{k}\cdot\hat{i} = 0 \end{equation*}
or,
\begin{equation*} \hat{i}\cdot\hat{i} =\hat{j}\cdot\hat{j} = \hat{k}\cdot\hat{k} =1 \end{equation*}
Properties:
  1. Commutative law: \(\quad \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\)
  2. Distributive law: \(\quad \vec{a} \cdot \left(\vec{b}+\vec{c} \right) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}\)

Subsubsection 2.1.2.2 Vector (or Cross) Product

The vector product of two vectors \(\vec{a}\) and \(\vec{b}\) is denoted by \(\vec{a}\times\vec{b}\) and is defined as
\begin{equation*} \vec{a} \times \vec{b} = \mid{\vec{a}}\mid \mid{\vec{b}}\mid sin\theta \hat {n} = absin\theta {\hat{n}}, \end{equation*}
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\text{,}\) and \({\hat{n}}\) is a unit vector perpendicular to the plane made by vectors \(\vec{a}\) and \(\vec{b}\text{,}\) and is representing the direction of \(\vec{a}\times\vec{b}\text{.}\) Here \(\vec{a}\text{,}\) \(\vec{b}\text{,}\) and \({\hat{n}}\) form a right handed system, as shown in Figure 2.1.4.(a).
(a)
(b)
(c)
Figure 2.1.4. Vector Product of Two Vectors

Geometrical Interpretation:.

Let \(\vec{a}\) and \(\vec{b}\) be the adjacent sides of the parallelogram, as shown in Figure 2.1.4.(b), then
\begin{equation*} \vec{a}\times\vec{b} = \vec{OA}\times\vec{OB} = (OA)(OB) \sin\theta \hat{n} = \left(OA\cdot{BM}\right){\hat{n}} = base \cdot height \quad \hat{n} \end{equation*}
= vector area of parallelogram OACB.
i.e., a vector product of two vectors is the vector area of a parallelogram constructed by these vectors.

Vector product in determinant form.

If \(\vec{a} = a_{1}\hat{i}+a_{2}\hat{j}+ a_{3}\hat{k}\) and \(\vec{b} = b_{1}\hat{i}+b_{2}\hat{j}+ b_{3}\hat{k}\) then
\begin{equation*} \vec{a}\times\vec{b} =\left( {a_{1}\hat{i}+a_{2}\hat{j}+ a_{3}\hat{k}}\right)\times \left( {b_{1}\hat{i}+b_{2}\hat{j}+ b_{3}\hat{k}}\right) \end{equation*}
\begin{equation*} =(a_{2}b_{3}-a_{3}b_{2})\hat{i}- (a_{1}b_{3}-a_{3}b_{1})\hat{j} + (a_{1}b_{2}-a_{2}b_{1})\hat{k} \end{equation*}
\begin{equation*} =\begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{bmatrix} \end{equation*}

Mnemonics:.

\begin{equation*} \hat{i}\times\hat{j}=\hat{k};\qquad \hat{j}\times\hat{k}=\hat{i};\qquad \hat{k}\times\hat{i}=\hat{j} \end{equation*}
or,
\begin{equation*} \hat{i}\times\hat{k}=-\hat{j};\qquad \hat{k}\times\hat{j}=-\hat{i};\qquad \hat{j}\times\hat{i}=-\hat{k} \end{equation*}

Properties:.

  1. Commutative Law: \(\quad \vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}\)
  2. Distributive law: \(\quad \vec{a} \times \left(\vec{b}+\vec{c} \right) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}\)