Vectors revision notes

• Triangle Law of Vector Addition: If a displacement goes from A to B ($\vec{AB}$), and then B to C ($\vec{BC}$), the net displacement is $\vec{AC} = \vec{AB} + \vec{BC}$. When the sides of a triangle are taken in order, their resultant is zero ($\vec{AB} + \vec{BC} + \vec{CA} = \vec{0}$).
• Parallelogram Law: If two vectors are represented by the two adjacent sides of a parallelogram, their sum is represented in magnitude and direction by the diagonal passing through their common initial point.

• Components of a Vector: Vectors can be resolved along axes using unit vectors $\hat{i}, \hat{j}, \hat{k}$. $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, where $x, y, z$ are scalar components and $x\hat{i}, y\hat{j}, z\hat{k}$ are vector components.
• Vector Joining Two Points: Directed from $P_1(x_1, y_1, z_1)$ to $P_2(x_2, y_2, z_2)$ is $\vec{P_1P_2} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$.

• Definition: The scalar product of non-zero vectors $\vec{a}$ and $\vec{b}$ is $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between them.
• Conditions: If $\vec{a}$ or $\vec{b}$ is zero, $\theta$ is undefined, and $\vec{a} \cdot \vec{b} = 0$. $\vec{a} \cdot \vec{b} = 0$ for non-zero vectors iff they are perpendicular.

• Orthogonal Unit Vectors: $\hat{i}\cdot\hat{i} = \hat{j}\cdot\hat{j} = \hat{k}\cdot\hat{k} = 1$ and $\hat{i}\cdot\hat{j} = \hat{j}\cdot\hat{k} = \hat{k}\cdot\hat{i} = 0$.
• Component Form: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$.

• Definition: The vector product is $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$. Here, $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$, forming a right-handed system (thumb points to $\hat{n}$ when fingers curl from $\vec{a}$ to $\vec{b}$).
• Conditions: If either vector is zero, $\vec{a} \times \vec{b} = \vec{0}$. For non-zero vectors, $\vec{a} \times \vec{b} = \vec{0}$ iff $\vec{a}$ and $\vec{b}$ are parallel/collinear.

• Orthogonal Unit Vectors: $\hat{i}\times\hat{i} = \hat{j}\times\hat{j} = \hat{k}\times\hat{k} = \vec{0}$. Following the right-hand rule: $\hat{i}\times\hat{j} = \hat{k}, \hat{j}\times\hat{k} = \hat{i}, \hat{k}\times\hat{i} = \hat{j}$.
• Determinant Form: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$.

(These concepts frequently appear in JEE Advanced coordinate geometry and vector calculus)

• Lagrange’s Identity: $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2$. This relates the dot and cross product magnitudes and is exceptionally powerful for simplifying complex expressions in JEE Advanced.

• Magnitude: $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$. (Unit: determined by physical context, e.g., meters).
• Unit Vector Direction: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$. (Unitless direction indicator).
• Collinearity proportionality: For $\vec{a} = \lambda \vec{b}$, $\frac{b_1}{a_1} = \frac{b_2}{a_2} = \frac{b_3}{a_3} = \lambda$.
• Angle formula (Dot Product): $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} = \frac{a_1b_1 + a_2b_2 + a_3b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}}$.
• Angle formula (Cross Product): $\sin\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}||\vec{b}|}$.

• Angle boundaries: The angle $\theta$ between vectors is strictly constrained to $0 \leq \theta \leq \pi$.
• Division by vectors: Mathematical division by a vector is undefined. Never write $\frac{\vec{a}}{\vec{b}}$. You cannot perform scalar algebraic "cancellation" of vectors across products.
• Zero Vector Denominators: Evaluating a unit vector $\hat{a}$ is impossible if $\vec{a} = \vec{0}$ because division by zero magnitude is undefined.
• Triangle Area Applicability: The formula $\frac{1}{2}|\vec{a} \times \vec{b}|$ requires vectors $\vec{a}$ and $\vec{b}$ to represent adjacent sides co-originating from a common vertex.

• Collinear vs Identical Line of Action: Students assume collinear vectors must physically lie on the same straight line. In vector algebra, we use "free vectors," so parallel vectors are mathematically classified as collinear regardless of their origin.
• Cross Product Commutativity Trap: Carelessly writing $\vec{a} \times \vec{b} = \vec{b} \times \vec{a}$ causes fatal sign errors. Reversing the order strictly reverses the direction via the right-hand rule: $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$.
• Direction Ratios vs Direction Cosines: Direction ratios $(a, b, c)$ can be any real numbers proportional to the vector's coordinates. Students falsely apply $a^2+b^2+c^2=1$. This sum is ONLY equal to 1 for normalized Direction Cosines ($l, m, n$).
• Triangle Law Head-to-Tail Traps: Placing vectors tail-to-tail to add them geometrically is incorrect for the Triangle Law (which requires head-to-tail). Tail-to-tail placement invokes the Parallelogram Law. When traversing a closed triangle sequentially, $\vec{AB} + \vec{BC} + \vec{CA} = \vec{0}$, NOT $\vec{AC}$.
• Scalar vs Vector Projections: Always note whether the question asks for the "projection" (a scalar length, $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$) or the "projection vector" (a vector quantity, $\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$).

• Orthogonal Projections: Finding vectors in the plane of two given vectors that are perpendicular to a third.
• Simultaneous Vector Equations: Solving for unknown vectors using systems like $\vec{x} \times \vec{a} = \vec{b}$ and $\vec{x} \cdot \vec{c} = d$ using the Vector Triple Product.
• Coplanarity Constraints: Using the Scalar Triple Product $= 0$ to find unknown constants in coordinate vertices.
• Area and Volume Geometry: Given diagonals (instead of sides), using $\frac{1}{2}|\vec{d_1} \times \vec{d_2}|$ to compute the area of a generic parallelogram.