The Chapter in One Sweep

A vector has magnitude and direction; a directed segment AB→\overrightarrow{AB} pictures it, and sliding it parallel to itself changes nothing (free vectors).

Directed segment from A to B and a position vector with direction angles

The vocabulary

Position vector OP→=xi^+yj^+zk^\overrightarrow{OP} = x\hat{i} + y\hat{j} + z\hat{k} with ∣OP→∣=x2+y2+z2\left|\overrightarrow{OP}\right| = \sqrt{x^2 + y^2 + z^2}. Direction cosines l=cos⁡α,m=cos⁡β,n=cos⁡γl = \cos\alpha, m = \cos\beta, n = \cos\gamma satisfy l2+m2+n2=1l^2 + m^2 + n^2 = 1; any proportional triple gives direction ratios — for a component-form vector, the components are direction ratios. Types: zero, unit (a^=a⃗/∣a⃗∣\hat{a} = \vec{a}/|\vec{a}|), coinitial, collinear (b⃗=λa⃗\vec{b} = \lambda\vec{a}, proportional components), equal (same magnitude and direction), negative.

Addition and scalar multiplication

Triangle law (tail-to-tip) and parallelogram law (common tail, diagonal) are the same fact; sides of a closed polygon taken in order sum to 0⃗\vec{0}; a⃗−b⃗=a⃗+(−b⃗)\vec{a} - \vec{b} = \vec{a} + \left(-\vec{b}\right); ∣λa⃗∣=∣λ∣∣a⃗∣|\lambda\vec{a}| = |\lambda||\vec{a}| with direction kept (λ>0\lambda > 0) or flipped (λ<0\lambda < 0).

Triangle law tail to tip and parallelogram law with diagonal sum

Joining points, dividing segments

P1P2→=\overrightarrow{P_1P_2} = terminal minus initial, componentwise. RR dividing PQPQ in m:nm : n: internally r⃗=mb⃗+na⃗m+n\vec{r} = \dfrac{m\vec{b} + n\vec{a}}{m + n}; externally r⃗=mb⃗−na⃗m−n\vec{r} = \dfrac{m\vec{b} - n\vec{a}}{m - n}; midpoint a⃗+b⃗2\dfrac{\vec{a} + \vec{b}}{2}; centroid of a triangle a⃗+b⃗+c⃗3\dfrac{\vec{a}+\vec{b}+\vec{c}}{3}.

Point R dividing PQ internally with position vector formula

The two products

Dot (scalar) Cross (vector)
Definition a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = \vert\vec{a}\vert\vert\vec{b}\vert\cos\theta a⃗×b⃗=∣a⃗∣∣b⃗∣sin⁡θ n^\vec{a}\times\vec{b} = \vert\vec{a}\vert\vert\vec{b}\vert\sin\theta\,\hat{n}
Output number vector ⊥\perp both (right-hand rule)
Components a1b1+a2b2+a3b3a_1b_1 + a_2b_2 + a_3b_3 3×33\times 3 determinant with i^,j^,k^\hat{i}, \hat{j}, \hat{k} on top
Zero means perpendicular parallel
Order commutative anticommutative: b⃗×a⃗=−a⃗×b⃗\vec{b}\times\vec{a} = -\vec{a}\times\vec{b}
Geometry projection =a⃗⋅b⃗∣b⃗∣= \dfrac{\vec{a}\cdot\vec{b}}{\vert\vec{b}\vert} area: parallelogram ∣a⃗×b⃗∣\vert\vec{a}\times\vec{b}\vert, triangle half of it

Dot product definition with angle and projection of a vector on a line

Cross product perpendicular to a parallelogram and the i j k cycle

Identities and the Mistake Checklist

The identity kit

  1. a⃗⋅a⃗=∣a⃗∣2\vec{a}\cdot\vec{a} = |\vec{a}|^2; ∣a⃗±b⃗∣2=∣a⃗∣2±2 a⃗⋅b⃗+∣b⃗∣2\left|\vec{a} \pm \vec{b}\right|^2 = |\vec{a}|^2 \pm 2\,\vec{a}\cdot\vec{b} + |\vec{b}|^2 — the workhorse conversions between lengths and dots.
  2. (a⃗+b⃗)⋅(a⃗−b⃗)=∣a⃗∣2−∣b⃗∣2\left(\vec{a}+\vec{b}\right)\cdot\left(\vec{a}-\vec{b}\right) = |\vec{a}|^2 - |\vec{b}|^2 (rhombus diagonals perpendicular; rectangle diagonals equal).
  3. Lagrange: ∣a⃗×b⃗∣2+(a⃗⋅b⃗)2=∣a⃗∣2∣b⃗∣2\left|\vec{a}\times\vec{b}\right|^2 + \left(\vec{a}\cdot\vec{b}\right)^2 = |\vec{a}|^2|\vec{b}|^2.
  4. Unit-vector half angles: ∣a^+b^∣=2cos⁡θ2\left|\hat{a}+\hat{b}\right| = 2\cos\dfrac{\theta}{2}, ∣a^−b^∣=2sin⁡θ2\left|\hat{a}-\hat{b}\right| = 2\sin\dfrac{\theta}{2}.
  5. Cycle: i^×j^=k^\hat{i}\times\hat{j} = \hat{k}, j^×k^=i^\hat{j}\times\hat{k} = \hat{i}, k^×i^=j^\hat{k}\times\hat{i} = \hat{j}; against the cycle, negative; self-cross zero. Dots: same pair 11, different pair 00.
  6. Inequalities: ∣a⃗⋅b⃗∣≤∣a⃗∣∣b⃗∣\left|\vec{a}\cdot\vec{b}\right| \leq |\vec{a}||\vec{b}| (Cauchy-Schwarz); ∣a⃗+b⃗∣≤∣a⃗∣+∣b⃗∣\left|\vec{a}+\vec{b}\right| \leq |\vec{a}| + |\vec{b}| (triangle).
  7. (JEE) Scalar triple product: [a⃗ b⃗ c⃗]=\left[\vec{a}\ \vec{b}\ \vec{c}\right] = determinant of components == box volume; zero   ⟺  \iff coplanar; swap flips sign, cycle preserves.

The mistake checklist — run it before submitting

  1. Joining vector backwards? Terminal minus initial. AB→\overrightarrow{AB} starts at AA.
  2. Section formula cross-paired? In mb⃗+na⃗m+n\dfrac{m\vec{b} + n\vec{a}}{m+n}, the first ratio number mm multiplies the far point QQ.
  3. The j^\hat{j}-cofactor minus sign in every cross-product determinant: expand "plus, MINUS, plus".
  4. Halved the triangle area? 12∣a⃗×b⃗∣\dfrac{1}{2}\left|\vec{a}\times\vec{b}\right| for a triangle; no half for a parallelogram; half of the diagonal cross for a parallelogram given by diagonals.
  5. Projection divided by the right magnitude? Projection of a⃗\vec{a} on b⃗\vec{b} divides by ∣b⃗∣|\vec{b}|.
  6. Normalise before scaling: a vector of magnitude MM along a⃗\vec{a} is M∣a⃗∣a⃗\dfrac{M}{|\vec{a}|}\vec{a} — never Ma⃗M\vec{a}.
  7. sin⁡θ\sin\theta ambiguity: cross-product data alone cannot separate θ\theta from π−θ\pi - \theta; check the dot's sign.
  8. Cancelling products: a⃗⋅b⃗=a⃗⋅c⃗\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c} or a⃗×b⃗=a⃗×c⃗\vec{a}\times\vec{b} = \vec{a}\times\vec{c} does NOT give b⃗=c⃗\vec{b} = \vec{c} — only that the difference is perpendicular (dot) or parallel (cross) to a⃗\vec{a}.

Where each skill was built

Vocabulary and direction cosines — Section 1. Addition laws and scalar multiplication — Section 2. Components, joining vector, section formula — Section 3. Dot product and projections — Section 4. Cross product and areas — Section 5. Graded worked examples — Section 6. Board-pattern practice — Section 7. JEE tools (triple products, identities, bisectors) — Section 8. Full JEE drill — Section 9.