# Algebraic Formula for class 12^{th}:

Algebra is key to unlock mathematics calculation. Algebra uses letters and symbols for representation of numbers. It has to deal with solutions of algebraic equations all over by its formula. This part has a great importance in mathematics from basics algebra upto the research level. Algebra is divided into two parts as Elementary algebra and Modern algebra (also known as Abstract algebra). We deal with only elementary algebra which contains basics of algebra. For the professionals who research in algebra section of mathematics studies abstract algebra they are also known as algebraists.

Elementary algebra deals with calculation and solution based equations. Elementary algebra is also subcategorized into different parts as vector algebra, rational algebra etc.

In class 12^{th} we have to study vector algebra in mathematics. As we know that the quantity which has both magnitude as well as direction is vector. Here also the same situation. To study linear algebra it’s important to study vector algebra in mathematics as well as vector quantity in physics it is necessary to note down the important points regarding algebra of vector positions. Solution of problems based on it will be easy to solve.

You very well know the importance of this chapter for board exam as well as every competitive exam.

## Here is some important notes regarding vector algebra:

- If \(\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}\) then magnitude or length or norm or absolute value of \(\vec{a}\) is \(\left | \overrightarrow{a} \right |=a=\sqrt{x^{2}+y^{2}+z^{2}}\)
- A vector of unit magnitude is unit vector. If \(\vec{a}\) is a vector then unit vector of \(\vec{a}\) is denoted as \(\hat{a}\) and \(\hat{a}=\frac{\hat{a}}{\left | \hat{a} \right |}\). Therefore \(\hat{a}=\frac{\hat{a}}{\left | \hat{a} \right |}\hat{a}\).
- Important unit vectors are \(\hat{i}, \hat{j}, \hat{k}\) , where \(\hat{i} = [1,0,0],\: \hat{j} = [0,1,0],\: \hat{k} = [0,0,1]\)
- If l=cosα, m=cosβ, n=cosγ, then α, β, γ, are called directional angles of the vector \(\overrightarrow{a}\) and cos2α+cos2β+cos2γ=1

\(\vec{a}+\vec{b}=\vec{b}+\vec{a}\) |

\(\vec{a}+\left ( \vec{b}+ \vec{c} \right )=\left ( \vec{a}+ \vec{b} \right )+\vec{c}\) |

\(k\left ( \vec{a}+\vec{b} \right )=k\vec{a}+k\vec{b}\) |

\(\vec{a}+\vec{0}=\vec{0}+\vec{a}\) therefore \(\vec{0}\) is the additive identity in vector addition. |

\(\vec{a}+\left ( -\vec{a} \right )=-\vec{a}+\vec{a}=\vec{0}\) therefore \(\vec{a}\) is the inverse in vector addition. |

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