Vector Algebra Cheatsheet

Cheatsheet Content

### Algebra of Vectors - **Definition:** A vector is a quantity having both magnitude and direction. Represented as $\vec{a}$ or $\mathbf{a}$. - **Position Vector:** The position vector of a point $P(x, y, z)$ from the origin $O(0, 0, 0)$ is $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$. - **Vector Addition (Triangle Law):** If $\vec{a}$ and $\vec{b}$ are two vectors, then $\vec{a} + \vec{b}$ is the third side of the triangle formed by $\vec{a}$ and $\vec{b}$. - Component form: If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then $\vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}$. - **Vector Subtraction:** $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$. - **Scalar Multiplication:** For a scalar $k$ and vector $\vec{a}$, $k\vec{a}$ is a vector with magnitude $|k||\vec{a}|$ and direction same as $\vec{a}$ if $k>0$, opposite if $k ### Vector Modulus - **Magnitude (Modulus) of a Vector:** For $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, its magnitude is $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$. - **Distance between two points:** For $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, the vector $\vec{AB} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$. - The distance is $|\vec{AB}| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. ### Scalar (Dot) Product - **Definition:** $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. - **Component form:** If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$. - **Properties:** - Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ - Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$ - $\vec{a} \cdot \vec{a} = |\vec{a}|^2$ - Orthogonality: If $\vec{a} \cdot \vec{b} = 0$ (and $\vec{a}, \vec{b}$ are non-zero), then $\vec{a} \perp \vec{b}$. - **Applications:** - **Angle between vectors:** $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$ - **Projection of $\vec{a}$ on $\vec{b}$:** $\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$ - **Scalar projection of $\vec{a}$ on $\vec{b}$:** $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ - **Work done by a force:** $W = \vec{F} \cdot \vec{d}$ ### Vector (Cross) Product - **Definition:** $\vec{a} \times \vec{b} = (|\vec{a}||\vec{b}|\sin\theta)\hat{n}$, where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ (right-hand rule). - **Component form:** $$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2b_3 - a_3b_2)\hat{i} - (a_1b_3 - a_3b_1)\hat{j} + (a_1b_2 - a_2b_1)\hat{k}$$ - **Properties:** - Anti-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$ - Distributive: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$ - Parallelism: If $\vec{a} \times \vec{b} = \vec{0}$ (and $\vec{a}, \vec{b}$ are non-zero), then $\vec{a} \parallel \vec{b}$. - $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$ - $\hat{j} \times \hat{i} = -\hat{k}$, etc. - **Applications:** - **Area of a parallelogram:** $|\vec{a} \times \vec{b}|$ (where $\vec{a}, \vec{b}$ are adjacent sides) - **Area of a triangle:** $\frac{1}{2}|\vec{a} \times \vec{b}|$ - **Moment (Torque):** $\vec{\tau} = \vec{r} \times \vec{F}$ - **Velocity of a point in a rotating body:** $\vec{v} = \vec{\omega} \times \vec{r}$ ### Scalar Triple Product (Box Product) - **Definition:** $\vec{a} \cdot (\vec{b} \times \vec{c})$ denoted as $[\vec{a}, \vec{b}, \vec{c}]$ or $(\vec{a} \ \vec{b} \ \vec{c})$. - **Component form:** $$[\vec{a}, \vec{b}, \vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$$ - **Properties:** - Cyclic permutation: $[\vec{a}, \vec{b}, \vec{c}] = [\vec{b}, \vec{c}, \vec{a}] = [\vec{c}, \vec{a}, \vec{b}]$ - Order change: $[\vec{a}, \vec{b}, \vec{c}] = -[\vec{b}, \vec{a}, \vec{c}]$ - Three vectors are coplanar if $[\vec{a}, \vec{b}, \vec{c}] = 0$. - **Applications:** - **Volume of a parallelepiped:** $|[\vec{a}, \vec{b}, \vec{c}]|$ (where $\vec{a}, \vec{b}, \vec{c}$ are co-terminus edges). - **Volume of a tetrahedron:** $\frac{1}{6}|[\vec{a}, \vec{b}, \vec{c}]|$. ### Vector Triple Product - **Definition:** $\vec{a} \times (\vec{b} \times \vec{c})$ - **Formula (BAC-CAB rule):** $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$ - **Properties:** - Not associative: $\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$ - Jacobi Identity: $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = \vec{0}$ - **Applications:** Used in physics for calculations involving angular momentum and electromagnetic theory.