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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 \(\vec B \).
\(\text{Projection of Vector} \vec {A} \ \text{on Vector} \vec{B} = \dfrac{\vec{A}. \vec{B}}{ \vec{B}}\)
Let us try out a few examples to easily understand the vector formulas.
Solved Examples on Vector Formulas

Example 1: Find the dot product of the vectors \(3\hat i 2\hat j + 7\hat k\) and \(4\hat i  \hat j + 3\hat k\).
Solution:
Given \(\vec A = 3\hat i  2\hat j + 7\hat k\) and \(\vec B = 4\hat i \hat j + 3\hat k\)
\( \vec A .\vec B = ((3).(4) + (2).(1) + 7.(3)) = 12 + 2 + 21 = 35\) 
Example 2: What is the angle between the vectors\(\vec A = \hat i + 5\hat j + 2\hat k\) and \(\vec B = 2\hat i \hat j  k\hat k\).
Solution:
Given \(\vec A = \hat i + 5\hat j + 2\hat k\) and \(\vec B = 2\hat i \hat j  k\hat k\)
\( \vec A .\vec B = ((1).(2) + (5).(1) + 2.(1)) = 2  5  2 = 5\)
\( \vec{A} = \sqrt{1^2 + 5^2 + 2^2} = \sqrt{30} \text{ and } \vec{B} = \sqrt{2^2 + (1)^2 + (1)^2} = \sqrt6\)
\(\begin{align}\theta& = Cos^{1}(\frac{\vec A.\vec B}{ \vec{A}.\vec{B}})\\& = Cos^{1}( \frac{5}{\sqrt{30}.\sqrt6}) \\&= Cos^{1}(\frac{\sqrt5}{6})\end{align} \)
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