Algebra is one of the major parts of Mathematics in which general symbols and letters are used to represent quantities and numbers in equations and formulae. The more basic parts of algebra are called elementary algebra and more abstract parts are called modern algebra or abstract algebra. Algebra is very important as it includes everything from elementary equation solving to the study of abstractions such as rings, groups and fields.

**Vector Algebra** is included in Class 12 mathematics syllabus as its importance is multi-fold. Vector Algebra deals with vectors - things that have both directions and magnitudes. It is important in both mathematics and physics. Learning vector algebra will help you in handling geometric transformations and it is very important in understanding Linear Algebra.

Algebra Formulas For Class 12 |

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 by \(\hat{a}\) 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 \( l=\cos \alpha, m=\cos \beta, n=\cos\gamma,\) then \( \alpha, \beta, \gamma,\) are called directional angles of the vectors\(\overrightarrow{a}\) and \(\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1\) |

In Vector Addition |

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

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

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

\(\vec{a}+\vec{0}=\vec{0}+\vec{a}\), 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. |