Vector Basics & Direction Cosines

  • Position Vector: The position vector of a point P(x,y,z)P(x, y, z) is r⃗=xi^+yj^+zk^\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}.
  • Magnitude: ∣r⃗∣=x2+y2+z2|\vec{r}| = \sqrt{x^2 + y^2 + z^2}.
  • Unit Vector: A vector of magnitude 1 in the direction of a⃗\vec{a} is given by a^=a⃗∣a⃗∣\hat{a} = \frac{\vec{a}}{|\vec{a}|}.
  • Direction Cosines: The cosines of the angles (α,β,γ\alpha, \beta, \gamma) a vector makes with the coordinate axes. l=cos⁡α=x∣r⃗∣,m=cos⁡β=y∣r⃗∣,n=cos⁡γ=z∣r⃗∣l = \cos\alpha = \frac{x}{|\vec{r}|}, \quad m = \cos\beta = \frac{y}{|\vec{r}|}, \quad n = \cos\gamma = \frac{z}{|\vec{r}|}
  • Fundamental Property: l2+m2+n2=1l^2 + m^2 + n^2 = 1.
  • Vector between Two Points: From A(x1,y1,z1)A(x_1, y_1, z_1) to B(x2,y2,z2)B(x_2, y_2, z_2) is AB⃗=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^\vec{AB} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}.

Algebra of Vectors & Section Formula

  • Collinearity: Two vectors a⃗\vec{a} and b⃗\vec{b} are collinear (parallel) if a⃗=λb⃗\vec{a} = \lambda\vec{b} for some scalar λ\lambda.
  • Internal Division: The position vector of point RR dividing line segment ABAB (with position vectors a⃗,b⃗\vec{a}, \vec{b}) internally in the ratio m:nm:n is: r⃗=mb⃗+na⃗m+n\vec{r} = \frac{m\vec{b} + n\vec{a}}{m + n}
  • External Division: r⃗=mb⃗−na⃗m−n\vec{r} = \frac{m\vec{b} - n\vec{a}}{m - n}
  • Midpoint Formula: r⃗=a⃗+b⃗2\vec{r} = \frac{\vec{a} + \vec{b}}{2}.

Scalar (Dot) Product

  • Definition: a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta.
  • Component Form: If a⃗=a1i^+a2j^+a3k^\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} and b⃗=b1i^+b2j^+b3k^\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}, then a⃗⋅b⃗=a1b1+a2b2+a3b3\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3.
  • Orthogonality Condition: a⃗⋅b⃗=0  ⟺  a⃗⊥b⃗\vec{a} \cdot \vec{b} = 0 \iff \vec{a} \perp \vec{b}.
  • Angle Between Vectors: cos⁡θ=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}.
  • Scalar Projection: The projection of a⃗\vec{a} on b⃗\vec{b} is a⃗⋅b⃗∣b⃗∣\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}.
  • Vector Projection: (a⃗⋅b⃗∣b⃗∣2)b⃗\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right) \vec{b}.

Vector (Cross) Product

  • Definition: a⃗×b⃗=∣a⃗∣∣b⃗∣sin⁡θ n^\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \, \hat{n}, where n^\hat{n} is a unit vector perpendicular to both a⃗\vec{a} and b⃗\vec{b} given by the right-hand rule.
  • Determinant Form: a⃗×b⃗=∣i^j^k^a1a2a3b1b2b3∣\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}
  • Collinearity Condition: a⃗×b⃗=0⃗  ⟺  a⃗∥b⃗\vec{a} \times \vec{b} = \vec{0} \iff \vec{a} \parallel \vec{b}.
  • Anti-commutative: a⃗×b⃗=−(b⃗×a⃗)\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a}).
  • Area of Parallelogram: ∣a⃗×b⃗∣|\vec{a} \times \vec{b}| (if a⃗,b⃗\vec{a}, \vec{b} are adjacent sides).
  • Area of Triangle: 12∣a⃗×b⃗∣\frac{1}{2} |\vec{a} \times \vec{b}| (if a⃗,b⃗\vec{a}, \vec{b} are adjacent sides).

Important Identities and Relations

Identity Name Formula Application
Lagrange's Identity ∣a⃗×b⃗∣2+(a⃗⋅b⃗)2=∣a⃗∣2∣b⃗∣2|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2 Relating dot and cross products with magnitudes.
Self Dot Product a⃗⋅a⃗=∣a⃗∣2\vec{a} \cdot \vec{a} = |\vec{a}|^2 Squaring a vector sum/difference equation.
Square of Sum ∣a⃗+b⃗∣2=∣a⃗∣2+∣b⃗∣2+2(a⃗⋅b⃗)|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) Expanding magnitudes involving multiple vectors.
Parallelogram Law ∣a⃗+b⃗∣2+∣a⃗−b⃗∣2=2(∣a⃗∣2+∣b⃗∣2)|\vec{a} + \vec{b}|^2 + |\vec{a} - \vec{b}|^2 = 2(|\vec{a}|^2 + |\vec{b}|^2) Relating the diagonals and sides of a parallelogram.

Important Exam Tips for Board Exams

  • Notation is Key: Always use arrows (a⃗\vec{a}) for vectors and caps (i^\hat{i}) for unit vectors. Forgetting these repeatedly can lead to mark deductions.
  • Show the Determinant: When asked to calculate a cross product, explicitly write out the 3×33 \times 3 determinant before expanding it. This clearly demonstrates your method to the examiner.
  • Projection Formulas: Don't confuse scalar projection with vector projection. Read the question carefully to see which one is being asked. Always specify "units" for scalar projection.
  • Orthogonal vs. Collinear: Memorize the conditions firmly. Dot product is zero for perpendicularity; cross product is zero (or components are proportional) for collinearity.

Important Exam Tips for JEE Main & Advanced

  • The Squaring Trick: Whenever a problem gives you a relation like a⃗+b⃗+c⃗=0⃗\vec{a} + \vec{b} + \vec{c} = \vec{0}, your immediate reflex should be to square both sides. This unlocks the dot products (a⃗⋅b⃗+…\vec{a} \cdot \vec{b} + \dots) via ∣a⃗+b⃗+c⃗∣2=0|\vec{a} + \vec{b} + \vec{c}|^2 = 0.
  • Exploit Lagrange's Identity: If you are given the angle, the magnitudes, and the dot product, but need the magnitude of the cross product, use Lagrange's identity. It bypasses calculating the actual vectors and saves critical time.
  • Geometric Intuition: Instead of brute-forcing algebra, visualize the vectors. If a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0, they form a right-angled triangle. Use simple trigonometry and Pythagorean theorems rather than heavy vector algebra to solve for side lengths or medians.
  • Eliminating Variables: In equations like x⃗×a⃗=b⃗\vec{x} \times \vec{a} = \vec{b}, taking the dot product of both sides with a⃗\vec{a} often yields useful constraints (e.g., b⃗⋅a⃗=0\vec{b} \cdot \vec{a} = 0), or taking the cross product with another vector simplifies the VTP.