Vector Algebra class 12 Formulas:
A quantity which has both magnitudes, as well as direction, is said to be vector quantity. It is denoted by an arrow pointing direction and length of its tail as the magnitude. Thus, It's symbol is \(\overrightarrow{A}\) and its magnitude is given as A. Thus, read vector algebra formulas further.
Position Vector and Magnitude:
We write position vector of any point P(x, y, z) as \(\overrightarrow{OP}\) (=\(\overrightarrow{r}\) ) = x\(\hat{k}\) + y\(\hat{k}\) + z\(\hat{k}\).
Also, its magnitude as \(\sqrt{x^{2} + y^{2} + z^{2}}\).
Direction Ratio:
Scalar components of any vector are its direction ratios and represent its projections along the respective axes. Thus, direction ratio and direction ratios are considered in vector algebra formula.
Relation between Direction Ratio and Direction Cosine:
The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as:
l = \(\frac{a}{r}\), m = \(\frac{b}{r}\), n = \(\frac{c}{r}\)
Triangle Formula:
The vector algebra formulas are absolute for triangles. The vector sum of the three sides of a triangle taken in order is \(\overrightarrow{0}\).
Dividing the line segment in given ratio:

Internally:
We write the position vector of a point R dividing a line segment in the ratio m : n joining the points P and Q whose position vectors are a and b respectively as:
\(\frac{n\overrightarrow{a} + m\overrightarrow{b}}{m + n}\)
2. Externally:
Similarly we can find position vector of R as
\(\frac{n\overrightarrow{a}  m\overrightarrow{b}}{m  n}\)
Scalar product:
For two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) also having angle θ between them, the scalar product is:
\(\overrightarrow{a}.\overrightarrow{b}\) = \(\overrightarrow{a}\)\(\overrightarrow{b}\)cosθ
Thus, Also written as cosθ = \(\frac{\overrightarrow{a}.\overrightarrow{b}}{\overrightarrow{a}\overrightarrow{b}}\)
Vector product:
The cross product of two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) also having angle θ between them is:
\(\overrightarrow{a}\) x \(\overrightarrow{b}\) = \(\overrightarrow{a}\)\(\overrightarrow{b}\) sinθ \(\hat{n}\)
Here, \(\hat{n}\) is a unit vector. Also, lies perpenducular the plane of \(\overrightarrow{a}\) and \(\overrightarrow{b}\). Then, we see that it is simple to denote these at same extent.
Properties of vector Algebra:
If we write vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) in its component form as
\(\overrightarrow{a}\) = a_{1}\(\hat{i}\) + a_{2}\(\hat{j}\) + a_{3}\(\hat{k}\)
\(\overrightarrow{b}\) = b_{1}\(\hat{i}\) + b_{2}\(\hat{j}\) + b_{3}\(\hat{k}\) also λ be any scalar.
Then vector algebra formulas are:
 \(\overrightarrow{a}\) + \(\overrightarrow{b}\) = (a_{1} + b_{1})\(\hat{i}\) + (a_{2} + b_{2})\(\hat{j}\) + (a_{3} + b_{3})\(\hat{k}\)
 λ \(\overrightarrow{a}\) = (λa_{1})\(\hat{i}\) + (λa_{2})\(\hat{j}\) + (λa_{3})\(\hat{k}\)
 \(\overrightarrow{a}\) . \(\overrightarrow{b}\) = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}
 \(\vec{a} \times \vec{b} = \begin{vmatrix} i & j & k \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{vmatrix}\)
Thus, All the formulas will help you to deal with problem solving assesment. In vector algebra class 12 Everything is there with full details.