Vector Formulas

Vector formulas provide a list of formulas, helpful for conducting numerous arithmetic operations on the same vector, and between two vectors. Vectors have both a scalar and a vector component and these vector formulas help in performing the numerous operations on vectors in a systematic and easy manner.

What are the List of Vector Formulas?

The list of vector formulas includes formulas performing the operations for a single vector and across the vectors. The formulas of direction ratios, direction cosines, the magnitude of a vector, unit vector are performed on the same vector. And the formulas of dot product, cross product, projection of vectors, are performed across two vectors. 

Formula 1
Direction ratios of a vector \(\vec A \) give the lengths of the vector in the x, y, z directions respectively.

The direction ratios of vector \(\vec A = a \hat i + b \hat j + c \hat k \)  is a, b, c respectively.

Formula 2
Direction cosine of vector \(\vec A \) is the cosecant of the angle made by the vector with the x, y, z, axes respectively.

The direction cosines of vector \(\vec A = a \hat i +b \hat j + c \hat k \) is:
\[\begin{align} l = Cos \alpha &= \frac{a}{\sqrt{a^2 + b^2 + c^2}}\\m = Cos \beta &= \frac{b}{\sqrt{a^2 + b^2 + c^2}} \\n = Cos \gamma &= \frac{c}{\sqrt{a^2 + b^2 + c^2}}\\  l^2 + m^2 + n^2 &= 1\end{align} \]

Formula 3
Magnitude of a vector \(\vec A \) is \(|\vec{A}| \).

For a vector \(\vec A = a \hat i + b \hat j + c \hat k \)  its magnitude is:
\[ |\vec{A}| = \sqrt{a_1^2 + b_1^2 + c_1^2}\]

Formula 4
Unit vector of \(\vec A \)  is \(\hat A \).

\[ \hat A = \frac{\vec A}{|\vec{A}|}\]

Formula 5
The two parallel vectors \(\vec A \) and \(\vec B\) and are related with the following formula, and \(\lambda \) is a numeric constant.

\[ \hat A = \frac{\vec A}{|\vec{A}|}\]

Formula 6
The angle between two vectors \(\vec A \) and \(\vec B \) is the cosecant of the angle between the two vectors.

\[ \begin{align}\vec A = a_1 \hat i +b_1\hat j +c_1\hat k  \text{ and } \vec B =a_2 \hat i +b_2\hat j +c_2\hat k \\ Cos\theta = \dfrac{|a_1.a_2 + b_1.b_2 + c_1.c_2|}{|\sqrt{a_1^2 + b_1^2 + c_1^2}|.|\sqrt{a_2^2 + b_2^2 + c_2^2}|}\end{align}\]

Formula 7
Dot product of \(\vec A\) and \(\vec B \) is a scalar product.

\[ \vec A. \vec B = |\vec{A}|.|B|.Cos\theta\]

Formula 8
Cross product between \(\vec A\) and \(\vec B \) is a vector product.

\[ \vec A \times \vec B = | \vec{A}|.|\vec{B}|.Sin \theta\]

Formula 9 
Dot product and cross product of unit vectors \(\hat i \), \(\hat j \), and \(\hat k \).

\[\begin{align} \hat i.\hat i= \hat j.\hat j = \hat k.\hat k = 1 \\ \hat i.\hat j =\hat j.\hat k=\hat k.\hat i = 0\end{align} \]
\[\begin{align} \hat i\times\hat i= \hat j \times \hat j = \hat k\times \hat k = 0 \\ \hat i \times \hat j =\hat k;\hat j \times \hat k=\hat i; \hat k \times \hat i = \hat j\end{align} \]

Formula 10
Projection of a vector \(\vec A \) on vector \(\vec B \).

Let us try out a few examples to easily understand the vector formulas.