10Class 12 Maths

Vector Algebra formulas

Dot product, cross product and vector formulas. These also underpin the next chapter on 3D geometry.

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Basics

Magnitude: $\displaystyle |\vec a| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
Unit vector: $\displaystyle \hat a = \frac{\vec a}{|\vec a|}$
Section formula: $\displaystyle \vec r = \frac{m\vec b + n\vec a}{m + n}$
Internal division.

Definition: $\displaystyle \vec a\cdot\vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$
Perpendicular vectors: $\displaystyle \vec a\cdot\vec b = 0$
Projection of a on b: $\displaystyle \frac{\vec a\cdot\vec b}{|\vec b|}$

Magnitude: $\displaystyle |\vec a\times\vec b| = |\vec a||\vec b|\sin\theta$
Component form: $\displaystyle \vec a\times\vec b = \begin{vmatrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$
Parallel vectors: $\displaystyle \vec a\times\vec b = \vec 0$
Area of a triangle: $\displaystyle \tfrac12|\vec a\times\vec b|$
a and b are two sides.
Area of a parallelogram: $\displaystyle |\vec a\times\vec b|$
a and b are adjacent sides. With its diagonals, it's half the magnitude.

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