Vector Algebra
Kuch quantities sirf magnitude rakhti hain (jaise mass, time) — ye scalars hain. Kuch ko magnitude ke saath direction bhi chahiye (jaise force, velocity) — ye vectors hain. Is chapter me vectors ki algebra padhenge.
1. Vector Basics
- Vector ko \(\vec{a}\) ya \(\overrightarrow{AB}\) likhte hain; magnitude \(|\vec{a}|\).
- Unit vector: magnitude \(1\) wala vector, \(\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}\).
- Zero vector \(\vec{0}\): magnitude \(0\), direction undefined.
- Standard unit vectors: \(\hat{i}, \hat{j}, \hat{k}\) (axes ke along).
Position vector of point \((x, y, z)\):
\[ \vec{r} = x\,\hat{i} + y\,\hat{j} + z\,\hat{k}, \qquad |\vec{r}| = \sqrt{x^{2} + y^{2} + z^{2}}. \]
2. Addition aur Scalar Multiplication
\(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) aur \(\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) ho to:
\[ \vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}. \]
Scalar \(\lambda\) se: \(\lambda\vec{a} = \lambda a_1\hat{i} + \lambda a_2\hat{j} + \lambda a_3\hat{k}\).
3. Dot (Scalar) Product
\[ \vec{a} \cdot \vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3. \] Result ek scalar hota hai.
- Angle: \(\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}\).
- Perpendicular vectors: \(\vec{a}\cdot\vec{b} = 0\).
- Projection of \(\vec{a}\) on \(\vec{b}\): \(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\).
\(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b} = 2\hat{i} - \hat{j} + \hat{k}\): \[ \vec{a}\cdot\vec{b} = (1)(2) + (2)(-1) + (3)(1) = 2 - 2 + 3 = 3. \]
4. Cross (Vector) Product
\[ \vec{a} \times \vec{b} = |\vec{a}|\,|\vec{b}|\sin\theta\ \hat{n}, \] result ek vector hota hai jo \(\vec{a}\) aur \(\vec{b}\) dono ke perpendicular hai.
Determinant se:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}. \]
- Parallel vectors: \(\vec{a}\times\vec{b} = \vec{0}\).
- \(|\vec{a}\times\vec{b}|\) = \(\vec{a}, \vec{b}\) se bane parallelogram ka area.
- Triangle ka area \(= \dfrac{1}{2}|\vec{a}\times\vec{b}|\).
Key Takeaways
- Vector = magnitude + direction; \(\vec{r} = x\hat{i}+y\hat{j}+z\hat{k}\), \(|\vec{r}| = \sqrt{x^{2}+y^{2}+z^{2}}\).
- Dot product = scalar; \(\vec{a}\cdot\vec{b} = 0\) ⇒ perpendicular.
- Cross product = vector (perpendicular to both); \(\vec{a}\times\vec{b} = \vec{0}\) ⇒ parallel.
- \(|\vec{a}\times\vec{b}|\) = parallelogram area; aadha = triangle area.