Vector Algebra

Vectors represent quantities with both magnitude and direction. They are fundamental in physics and mathematics.

Overview

graph TD
    A[Vectors] --> B[Basic Operations]
    A --> C[Products]
    A --> D[Applications]
    C --> C1[Dot Product]
    C --> C2[Cross Product]

Vector Basics

Representation

$$\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$$

Magnitude

$$|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$$

Unit Vector

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

Position Vector

$$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

Vector Operations

Addition

$$\vec{a} + \vec{b} = (a_x+b_x)\hat{i} + (a_y+b_y)\hat{j} + (a_z+b_z)\hat{k}$$

Triangle Law: Head-to-tail addition

Parallelogram Law: Diagonal of parallelogram

Scalar Multiplication

$$k\vec{a} = ka_x\hat{i} + ka_y\hat{j} + ka_z\hat{k}$$

Dot Product (Scalar Product)

$$\boxed{\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_xb_x + a_yb_y + a_zb_z}$$

Properties

1. Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
2. Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
3. $\vec{a} \cdot \vec{a} = |\vec{a}|^2$
4. Perpendicular: $\vec{a} \cdot \vec{b} = 0$

Angle Between Vectors

$$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$$

Projection

Projection of $\vec{a}$ on $\vec{b}$:

$$\text{Scalar} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$$ $$\text{Vector} = \frac{(\vec{a} \cdot \vec{b})}{|\vec{b}|^2}\vec{b}$$

Cross Product (Vector Product)

$$\boxed{\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n}}$$ $$= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}$$

Properties

1. Anti-commutative: $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$
2. Distributive: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$
3. $\vec{a} \times \vec{a} = \vec{0}$
4. Parallel: $\vec{a} \times \vec{b} = \vec{0}$

Magnitude

$$|\vec{a} \times \vec{b}| = \text{Area of parallelogram}$$

Scalar Triple Product

$$[\vec{a}, \vec{b}, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}$$

• Equals volume of parallelepiped
• Zero if vectors are coplanar

Vector Triple Product

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

Memory: “BAC - CAB” rule

Applications

Collinear Vectors

$\vec{a}$ and $\vec{b}$ are collinear if $\vec{a} = \lambda\vec{b}$

Coplanar Vectors

Three vectors are coplanar if $[\vec{a}, \vec{b}, \vec{c}] = 0$

Area of Triangle

$$\text{Area} = \frac{1}{2}|\vec{a} \times \vec{b}|$$

Practice Problems

1. Find the angle between vectors $2\hat{i} + \hat{j}$ and $\hat{i} + 2\hat{j}$.

2. Find a unit vector perpendicular to both $\vec{a} = \hat{i} + \hat{j}$ and $\vec{b} = \hat{j} + \hat{k}$.

3. Find the volume of parallelepiped with edges $\hat{i}+\hat{j}$, $\hat{j}+\hat{k}$, $\hat{k}+\hat{i}$.

Further Reading

• 3D Geometry - Lines and planes
• Coordinate Geometry - 2D applications