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Vectors (Mathematics) revision notes

Every idea in the Vectors (Mathematics) chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 24 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Vectors, position vectors and section formula

Magnitude and types of vectors
4× asked, last 2024
a⃗=a1i^+a2j^+a3k^,∣a⃗∣=a12+a22+a32,a^=a⃗∣a⃗∣\vec a=a_1\hat i+a_2\hat j+a_3\hat k,\qquad |\vec a|=\sqrt{a_1^2+a_2^2+a_3^2},\qquad \hat a=\frac{\vec a}{|\vec a|}
Zero vector: no direction. Equal: same length and direction. Collinear: b⃗=ka⃗\vec b=k\vec a. Coplanar: in one plane.
  • Not every root after squaring. a⃗⋅v⃗/∣v⃗∣=3\vec a\cdot\vec v/|\vec v|=3 needs a⃗⋅v⃗>0\vec a\cdot\vec v>0. Square to solve, then keep only the roots that satisfy the unsquared equation.
Position vectors and section formula
5× asked, last 2026
AB→=b⃗−a⃗,p⃗=mb⃗+na⃗m+n  (AP:PB=m:n)\overrightarrow{AB}=\vec b-\vec a,\qquad \vec p=\frac{m\vec b+n\vec a}{m+n}\ \ (AP:PB=m:n)
Internal division; external: (mb⃗−na⃗)/(m−n)(m\vec b-n\vec a)/(m-n). Midpoint (a⃗+b⃗)/2(\vec a+\vec b)/2; centroid (a⃗+b⃗+c⃗)/3(\vec a+\vec b+\vec c)/3.
a⃗\vec ab⃗\vec bb⃗−a⃗\vec b-\vec aAABBPP: 2:1 insideQQ: 2:1 outsideOO
AB→=b⃗−a⃗\overrightarrow{AB}=\vec b-\vec a (head minus tail); P=2b⃗+a⃗3P=\frac{2\vec b+\vec a}3, Q=2b⃗−a⃗1Q=\frac{2\vec b-\vec a}{1}
  • Centroid GG: GA2+GB2+GC2=(AB2+BC2+CA2)/3GA^2+GB^2+GC^2=(AB^2+BC^2+CA^2)/3. The median from AA has length ∣(b⃗+c⃗)/2−a⃗∣|(\vec b+\vec c)/2-\vec a|.
  • Not tail minus head. PM→=m⃗−p⃗\overrightarrow{PM}=\vec m-\vec p (head minus tail). Writing p⃗−m⃗\vec p-\vec m gives MP→\overrightarrow{MP}: the same length with every sign flipped.
Angle bisector meets the opposite side
3× asked, last 2024
BDDC=ABAC,d⃗=∣AC∣ b⃗+∣AB∣ c⃗∣AB∣+∣AC∣\frac{BD}{DC}=\frac{AB}{AC},\qquad \vec d=\frac{|AC|\,\vec b+|AB|\,\vec c}{|AB|+|AC|}
Internal bisector of angle AA meets BCBC at DD. Find both side lengths first; if AB=ACAB=AC, DD is the midpoint.
CCOOAABB3612
∣OA∣=3|OA|=3, ∣OB∣=6|OB|=6: the bisector cuts ABAB at CC with AC:CB=1:2AC:CB=1:2
  • Not weighting each end by its own part. CC on ABAB with AC:CB=1:2AC:CB=1:2 is (2a⃗+b⃗)/3(2\vec a+\vec b)/3. Example: ∣OA∣=3|OA|=3, ∣OB∣=6|OB|=6, so the bisector of ∠AOB\angle AOB cuts ABAB in the ratio 1:21:2.
Triangle centres
3× asked, last 2025
O (circumcentre) at origin:h⃗=a⃗+b⃗+c⃗,g⃗=h⃗+2o⃗3O\ \text{(circumcentre) at origin}:\quad \vec h=\vec a+\vec b+\vec c,\qquad \vec g=\frac{\vec h+2\vec o}{3}
GG divides HOHO in the ratio 2:12:1 (Euler line). In an equilateral triangle all the centres coincide.
OOGGHHAABBCC
Circumcentre OO at the origin: h⃗=a⃗+b⃗+c⃗\vec h=\vec a+\vec b+\vec c, GG on OHOH with HG:GO=2:1HG:GO=2:1 (Euler line)
  • Not 2PQ→2\overrightarrow{PQ}. From the circumcentre PP, PA→+PB→+PC→=3PG→=PQ→\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=3\overrightarrow{PG}=\overrightarrow{PQ} (QQ orthocentre). From QQ the sum is 2QP→2\overrightarrow{QP}.
Collinear points
4× asked, last 2024
A,B,C collinear  ⟺  AB→=λ AC→  ⟺  AB→×AC→=0⃗A,B,C\ \text{collinear}\iff\overrightarrow{AB}=\lambda\,\overrightarrow{AC}\iff\overrightarrow{AB}\times\overrightarrow{AC}=\vec 0
Equate components, starting with one that has no unknown: it gives λ\lambda. Collinear vectors: b⃗=λa⃗\vec b=\lambda\vec a.

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19 more concepts in 4 topics

Dot product, Cross product, Solving vector equations and Scalar triple product and coplanarity. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Dot product

  • Dot product and angle13×
  • Acute or obtuse angle4×
  • Projection and resolving a vector23×
  • Squares of sums and differences14×
  • Unit vectors adding to zero3×
  • Bounds and greatest values3×
  • Angle bisector direction4×
  • Rotating a vector5×

Cross product

  • Cross product10×
  • Mixing a, b and a × b12×
  • Parallel vectors and the common normal11×
  • Lagrange's identity13×
  • Areas19×
  • Cross product taken twice19×

Solving vector equations

  • Same vector crossed on both sides26×
  • Given a × x and a · x13×
  • Vector in the plane of a and b11×
  • Coefficients of a linear combination8×

Scalar triple product and coplanarity

  • Scalar triple product [a b c]31×

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