Vector Algebra — JEE Mains Mathematics

Formulas (3)

• Angle between two vectors; perpendicular, acute and obtuse · Angle from components
  $$\cos\theta=\frac{a_1b_1+a_2b_2+a_3b_3}{|\vec a|\,|\vec b|}$$
• Projections and components along and across a vector · Scalar projection
  $$\text{proj}_{\vec b}\vec a=\frac{\vec a\cdot\vec b}{|\vec b|}$$
• A vector in the plane of two others · In the plane of a and b
  $$\vec v=x\,\vec a+y\,\vec b$$

Watch out for (3)

• A negative dot product includes π
  → Angle between two vectors; perpendicular, acute and obtuse
• |b| for the length, |b|² for the vector
  → Projections and components along and across a vector
• The direction is not the vector
  → A vector in the plane of two others

Formulas (2)

• Expanding the square of a sum · Square of a sum
  $$|\vec a+\vec b|^2=|\vec a|^2+2\,\vec a\cdot\vec b+|\vec b|^2$$
• Combinations with a cross-product term · Length with a cross term (unit vectors)
  $$|\lambda\vec a+\mu\vec b+\nu(\vec a\times\vec b)|^2=\lambda^2+\mu^2+2\lambda\mu\cos\theta+\nu^2\sin^2\theta$$

Watch out for (2)

• Square before you add
  → Expanding the square of a sum
• Only for unit vectors
  → Combinations with a cross-product term

Formulas (3)

• Areas of triangles, parallelograms and quadrilaterals · Triangle area
  $$\text{Area}=\tfrac12\,|\overrightarrow{AB}\times\overrightarrow{AC}|$$
• A vector perpendicular to two others · Common perpendicular
  $$\vec c\perp\vec a,\ \vec c\perp\vec b\ \Rightarrow\ \vec c=\lambda(\vec a\times\vec b)$$
• |a × b|² + (a · b)² = |a|²|b|² · Lagrange's identity
  $$|\vec a\times\vec b|^2+(\vec a\cdot\vec b)^2=|\vec a|^2|\vec b|^2$$

Watch out for (3)

• Diagonals give half
  → Areas of triangles, parallelograms and quadrilaterals
• Two unit vectors, not one
  → A vector perpendicular to two others
• The identity is in squares
  → |a × b|² + (a · b)² = |a|²|b|²

Formulas (2)

• r × a = b × a: the difference is parallel to a · The parallel form
  $$\vec r\times\vec a=\vec b\times\vec a\ \Rightarrow\ \vec r=\vec b+\lambda\vec a$$
• a × c = b with a · c given: cross again with a · Solving a × c = b
  $$\vec c=\frac{(\vec a\cdot\vec c)\,\vec a-\vec a\times\vec b}{|\vec a|^2}$$

Watch out for (2)

• c × b is minus b × c
  → r × a = b × a: the difference is parallel to a
• Check solvability first
  → a × c = b with a · c given: cross again with a

Formulas (2)

• Scalar triple product: volume and coplanarity · Coplanarity
  $$[\vec a\ \vec b\ \vec c]=\vec a\cdot(\vec b\times\vec c)=0$$
• Vector triple product: a × (b × c) · BAC − CAB
  $$\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\,\vec b-(\vec a\cdot\vec b)\,\vec c$$

Watch out for (2)

• Points need edges, not positions
  → Scalar triple product: volume and coplanarity
• Not associative
  → Vector triple product: a × (b × c)

Formulas (3)

• Section points, centroid and collinearity · Section formula
  $$\vec p=\frac{n\,\vec a+m\,\vec b}{m+n}$$
• Triangle centres and angle bisectors · Internal bisector
  $$\text{direction}=\frac{\vec a}{|\vec a|}+\frac{\vec b}{|\vec b|}$$
• Rotating a vector, and rotating the axes · Plane rotation
  $$(x,y)\mapsto(x\cos\theta-y\sin\theta,\ x\sin\theta+y\cos\theta)$$

Watch out for (3)

• Cross the weights
  → Section points, centroid and collinearity
• Add unit vectors
  → Triangle centres and angle bisectors
• Which way it turns
  → Rotating a vector, and rotating the axes