Vectors

1. Algebra of Vectors

• Unit Vector

  $\hat{a} = \frac{\overline{a}}{|\overline{a}|}$

• Triangle Law of Addition

  $\overline{AB} + \overline{BC} = \overline{AC}$

  Triangle Law of Addition: $\overline{a} + \overline{b} = \overline{AC}$
• Vector Joining Two Points

  If $A \equiv (x_1, y_1, z_1)$ and $B \equiv (x_2, y_2, z_2)$, then

  $$ \overline{AB} = \overline{b} - \overline{a} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k} $$

  Position vectors and the vector joining two points A and B
• Collinearity

  $\overline{a}$ and $\overline{b}$ are collinear $\iff \overline{a} = t\overline{b}$ or $m\overline{a} + n\overline{b} = \overline{0}$.

• Coplanarity of 3 Vectors

  $\overline{a}$, $\overline{b}$, and $\overline{c}$ are coplanar $\iff x\overline{a} + y\overline{b} + z\overline{c} = \overline{0}$ for some scalars $x, y, z$ (not all zero).

2. 3D Coordinate System

• Distance Formulas

  Distance from Origin: $|\overline{OP}| = \sqrt{x^2 + y^2 + z^2}$

  Distance Between Two Points: $l(AB) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

  Distance from Coordinate Axes:

  From X-axis = $\sqrt{y^2 + z^2}$, Y-axis = $\sqrt{x^2 + z^2}$, Z-axis = $\sqrt{x^2 + y^2}$.

• 3D Coordinate Axes and Planes
  The 3D Coordinate Axes and Planes
• Signs of Coordinates in Different Octants

  Octant	I	II	III	IV	V	VI	VII	VIII
  Signs (x,y,z)	$(+,+,+)$	$(-,+,+)$	$(-,-,+)$	$(+,-,+)$	$(+,+,-)$	$(-,+,-)$	$(-,-,-)$	$(+,-,-)$

3. Section Formulas

• Internal and External Division

  Let $A(\overline{a})$ and $B(\overline{b})$ be two points divided by point $R(\overline{r})$.

  Internal Division (ratio $m:n$): $\overline{r} = \frac{m\overline{b} + n\overline{a}}{m + n}$

  External Division (ratio $m:n$): $\overline{r} = \frac{m\overline{b} - n\overline{a}}{m - n}$

• Midpoints & Centroids

  Midpoint Formula: $\overline{m} = \frac{\overline{a} + \overline{b}}{2}$

  Centroid of Triangle ABC: $\overline{g} = \frac{\overline{a} + \overline{b} + \overline{c}}{3}$

  Centroid of Tetrahedron ABCD: $\overline{g} = \frac{\overline{a} + \overline{b} + \overline{c} + \overline{d}}{4}$

  Incenter of Triangle ABC: $\overline{h} = \frac{|\overline{BC}|\overline{a} + |\overline{AC}|\overline{b} + |\overline{AB}|\overline{c}}{|\overline{BC}| + |\overline{AC}| + |\overline{AB}|}$

4. Dot Product (Scalar Product)

• Definition & Angles

  Definition: $\overline{a} \cdot \overline{b} = |\overline{a}| |\overline{b}| \cos \theta$

  Perpendicular Condition: $\overline{a} \cdot \overline{b} = 0 \iff \theta = \frac{\pi}{2}$

  Angle Between Vectors:

  $$ \cos \theta = \frac{\overline{a} \cdot \overline{b}}{|\overline{a}||\overline{b}|} = \frac{a_1b_1 + a_2b_2 + a_3b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}} $$

• Projections
  Angle $\theta$ and Projection

  Scalar Projection of $\overline{b}$ on $\overline{a}$: $\frac{\overline{a} \cdot \overline{b}}{|\overline{a}|}$

  Vector Projection of $\overline{b}$ on $\overline{a}$: $\left(\frac{\overline{a} \cdot \overline{b}}{|\overline{a}|^2}\right)\overline{a}$

• Direction Angles and Direction Cosines
  Direction angles mapped to X, Y, Z axes

  Direction Cosines ($l, m, n$): $l = \cos \alpha$, $m = \cos \beta$, $n = \cos \gamma$

  Relation: $l^2 + m^2 + n^2 = 1 \implies \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$

  Relation to Direction Ratios ($a, b, c$):

  $$ l = \pm\frac{a}{\sqrt{a^2+b^2+c^2}}, \quad m = \pm\frac{b}{\sqrt{a^2+b^2+c^2}}, \quad n = \pm\frac{c}{\sqrt{a^2+b^2+c^2}} $$

5. Cross Product (Vector Product)

• Cross Product Properties
  Right-Hand Rule and Cross Product direction

  Definition: $\overline{a} \times \overline{b} = |\overline{a}| |\overline{b}| \sin \theta \hat{n}$

  Parallel Condition: $\overline{a} \times \overline{b} = \overline{0} \iff \theta = 0$

  Unit Vector Perpendicular to Plane: $\hat{n} = \frac{\overline{a} \times \overline{b}}{|\overline{a} \times \overline{b}|}$

  Determinant Form:

  $$ \overline{a} \times \overline{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} $$

  Area of Parallelogram: $|\overline{a} \times \overline{b}|$

  Area of Triangle: $\frac{1}{2}|\overline{a} \times \overline{b}|$

6. Triple Products

• Scalar Triple Product (Box Product)
  Volume of a Parallelepiped

  Definition: $[\overline{a} \ \overline{b} \ \overline{c}] = \overline{a} \cdot (\overline{b} \times \overline{c})$

  Coplanarity Condition: $[\overline{a} \ \overline{b} \ \overline{c}] = 0 \iff$ vectors are coplanar.

  Volume of Parallelepiped: $V = |[\overline{a} \ \overline{b} \ \overline{c}]|$

  Volume of Tetrahedron: $V = \frac{1}{6}|[\overline{a} \ \overline{b} \ \overline{c}]|$

  Determinant Form:

  $$ [\overline{a} \ \overline{b} \ \overline{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} $$

• Vector Triple Product

  $$ \overline{a} \times (\overline{b} \times \overline{c}) = (\overline{a} \cdot \overline{c})\overline{b} - (\overline{a} \cdot \overline{b})\overline{c} $$