Vectors revision notes

Key Concepts & Definitions

Scalars

Quantities that involve only a magnitude (a real number). Examples: length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance.

Vectors

Quantities that have both magnitude and a specific direction. Examples: displacement, velocity, acceleration, force, weight, momentum, electric field intensity.

Directed Line Segment

A straight line segment with an arrowhead prescribing one direction. It has an initial point (start) and a terminal point (end), and acts as a geometric representation of a vector AB⃗\vec{AB}AB. The distance between these points is its magnitude, denoted as ∣AB⃗∣|\vec{AB}|∣AB∣ or ∣a⃗∣|\vec{a}|∣a∣.

Position Vector

For a point P(x,y,z)P(x,y,z)P(x,y,z) in a 3D right-handed rectangular coordinate system with origin O(0,0,0)O(0,0,0)O(0,0,0), the vector OP⃗\vec{OP}OP is the position vector r⃗\vec{r}r of point P. Its magnitude is given by the distance formula ∣r⃗∣=x2+y2+z2|\vec{r}| = \sqrt{x^2+y^2+z^2}∣r∣=x2+y2+z2​.

Direction Cosines & Ratios

A vector r⃗\vec{r}r makes direction angles α,β,γ\alpha, \beta, \gammaα,β,γ with the positive x,y,zx, y, zx,y,z axes respectively. Their cosines (l=cos⁡α,m=cos⁡β,n=cos⁡γl = \cos\alpha, m = \cos\beta, n = \cos\gammal=cosα,m=cosβ,n=cosγ) are the direction cosines. Any numbers a,b,ca, b, ca,b,c proportional to the direction cosines are called direction ratios.JEE TIPThe relation l2+m2+n2=1l^2+m^2+n^2=1l2+m2+n2=1 is strictly true, but in general, a2+b2+c2≠1a^2+b^2+c^2 \neq 1a2+b2+c2=1.

Types of Vectors

Zero (Null) Vector (0⃗\vec{0}0): Initial and terminal points coincide. It has zero magnitude and an arbitrary/undefined direction. Unit Vector: A vector with a magnitude of unity (1 unit). The unit vector in the direction of a⃗\vec{a}a is denoted as a^\hat{a}a^. Coinitial Vectors: Two or more vectors starting from the exact same initial point. Collinear Vectors: Two or more vectors that are parallel to the same line, irrespective of their individual magnitudes or directions. Equal Vectors: Two vectors a⃗\vec{a}a and b⃗\vec{b}b with identical magnitude and direction, regardless of where their initial points are located. Negative of a Vector: Same magnitude as the given vector, but the exact opposite direction (BA⃗=−AB⃗\vec{BA} = -\vec{AB}BA=−AB). Free Vectors: Vectors that can be subjected to parallel displacement without altering their magnitude and direction.

Addition of Vectors & Section Formula

• 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.
• Properties of Vector Addition:
  • Commutativity: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$.
  • Associativity: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$.
  • Additive Identity: $\vec{a} + \vec{0} = \vec{a}$.
• 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 $\vec{P_1P_2} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$.
• Section Formula: Let points P and Q have position vectors $\vec{a}$ and $\vec{b}$. A point R dividing the line segment PQ in ratio $m:n$ has the position vector:
  • Internally: $\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$.
  • Externally: $\vec{r} = \frac{m\vec{b} - n\vec{a}}{m-n}$.
  • Midpoint: $\vec{r} = \frac{\vec{a} + \vec{b}}{2}$.

Scalar (Dot) Product & Projections

• 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.
• Properties:
  • Commutativity: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$.
  • Distributivity: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$.
  • Scalar Multiplication: $(\lambda\vec{a}) \cdot \vec{b} = \lambda(\vec{a} \cdot \vec{b}) = \vec{a} \cdot (\lambda\vec{b})$.
• 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: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$.
• Projection:
  • The projection (scalar) of $\vec{a}$ on a directed line $l$ with unit vector $\hat{p}$ is $\vec{a} \cdot \hat{p}$.
  • The projection of $\vec{a}$ on $\vec{b}$ is $\vec{a} \cdot \hat{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.
  • The projection vector of $\vec{a}$ on $\vec{b}$ is $\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$.
• Inequalities:
  • Cauchy-Schwarz Inequality: $|\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}|$.
  • Triangle Inequality: $|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|$.JEE TIPEquality holds only if vectors are collinear and have the same direction.

Vector (Cross) Product & Areas

• 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.
• Properties:
  • Non-Commutative: $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$.
  • Distributivity: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$.
• 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: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$.
• Geometrical Applications (Areas):
  • Area of a Triangle with adjacent sides $\vec{a}$ and $\vec{b}$ is $\frac{1}{2}|\vec{a} \times \vec{b}|$.
  • Area of a Parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ is $|\vec{a} \times \vec{b}|$.

Advanced Vector Products (JEE Focus)

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

• Scalar Triple Product (STP): Denoted as $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$.
  • Geometrical Meaning: Represents the volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$. Volume of a tetrahedron is $\frac{1}{6}|[\vec{a}\vec{b}\vec{c}]|$.
  • Coplanarity: Three vectors are strictly coplanar if and only if $[\vec{a} \vec{b} \vec{c}] = 0$.
  • Cyclic Property: $[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}]$. Swapping any two adjacent vectors flips the sign: $[\vec{a} \vec{b} \vec{c}] = -[\vec{b} \vec{a} \vec{c}]$.
• Vector Triple Product (VTP): $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$.
  • JEE TIPMemorize this as the "BAC - CAB" rule. The resulting vector lies entirely in the plane formed by $\vec{b}$ and $\vec{c}$, and is perpendicular to $\vec{a}$.
• 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.

Conditions & Limitations

• 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.

Previous Year JEE Topics

• 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.

Top 10 JEE MCQ Traps

1. Misconception: $l^2+m^2+n^2 = 1$ applies to ANY scalar components $x, y, z$ of a vector $\rightarrow$ Correct Understanding: It strictly applies ONLY to direction cosines ($l, m, n$) which are derived from a unit vector.
2. Misconception: If $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, then $\vec{b} = \vec{c}$ $\rightarrow$ Correct Understanding: You cannot cancel $\vec{a}$. The equation actually means $\vec{a} \cdot (\vec{b}-\vec{c}) = 0$. Thus, either $\vec{a}=\vec{0}$, $\vec{b}=\vec{c}$, OR $\vec{a}$ is strictly perpendicular to the vector $(\vec{b}-\vec{c})$.
3. Misconception: If $\vec{a} \times \vec{b} = \vec{0}$, then either $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$ $\rightarrow$ Correct Understanding: This also happens if the non-zero vectors are parallel/collinear ($\theta = 0^\circ$ or $180^\circ$).
4. Misconception: The Vector Cross Product is associative: $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \times \vec{c}$ $\rightarrow$ Correct Understanding: Cross product is NOT associative. These represent entirely different planes in space (use the BAC-CAB rule to expand).
5. Misconception: The area of a parallelogram is always $|\vec{u} \times \vec{v}|$ $\rightarrow$ Correct Understanding: This is true if $\vec{u}$ and $\vec{v}$ are adjacent sides. If $\vec{u}$ and $\vec{v}$ are the diagonals, the area is $\frac{1}{2}|\vec{u} \times \vec{v}|$.
6. Misconception: The projection of $\vec{A}$ on $\vec{B}$ has $|\vec{A}|$ in the denominator $\rightarrow$ Correct Understanding: The vector being projected onto forms the denominator and direction. Projection is $\frac{\vec{A} \cdot \vec{B}}{|\vec{B}|}$.
7. Misconception: When finding the angle between two vectors geometrically, you can use the angle formed when they are arranged head-to-tail $\rightarrow$ Correct Understanding: The angle $\theta$ is strictly defined as the angle between their divergent paths when placed tail-to-tail (coinitial).
8. Misconception: If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$, the vectors are collinear $\rightarrow$ Correct Understanding: Squaring both sides shows $\vec{a} \cdot \vec{b} = -\vec{a} \cdot \vec{b}$, meaning $\vec{a} \cdot \vec{b} = 0$. The vectors are perpendicular, not collinear.
9. Misconception: Volume of a tetrahedron is the same as the Scalar Triple Product $|[\vec{a}\vec{b}\vec{c}]|$ $\rightarrow$ Correct Understanding: The STP gives the volume of a parallelepiped. For a tetrahedron defined by adjacent edges $\vec{a}, \vec{b}, \vec{c}$, the volume is exactly $\frac{1}{6} |[\vec{a}\vec{b}\vec{c}]|$.
10. Misconception: $(\vec{a} \cdot \vec{b})^2$ is equal to $\vec{a}^2 \vec{b}^2$ $\rightarrow$ Correct Understanding: Vectors cannot be algebraically squared like scalars in cross-multiplication. By Lagrange's Identity, $(\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 - |\vec{a} \times \vec{b}|^2$.