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Formulas: vectors, grade 12 (5 units)

Solids in space

  • The volume of a prism and a cylinder – S\displaystyle S the base’s area, h\displaystyle h the solid’s height
    V\displaystyle V =\displaystyle = S⋅h\displaystyle S\cdot h
  • The volume of a pyramid and a cone – S\displaystyle S the base’s area, h\displaystyle h the solid’s height
    V\displaystyle V =\displaystyle = S⋅h3\displaystyle \frac{S\cdot h}{3}
  • The lateral area of a right cone – R\displaystyle R the base’s radius, ℓ\displaystyle \ell the slant height
    M\displaystyle M =\displaystyle = πRℓ\displaystyle \pi R\ell
  • The volume of a sphere – R\displaystyle R its radius
    V\displaystyle V =\displaystyle = 43πR3\displaystyle \frac{4}{3}\pi R^3
  • The surface area of a sphere
    M\displaystyle M =\displaystyle = 4πR2\displaystyle 4\pi R^2
  • The diagonal of a box of dimensions a\displaystyle a, b\displaystyle b, c\displaystyle c
    d\displaystyle d =\displaystyle = a2+b2+c2\displaystyle \sqrt{a^2+b^2+c^2}

Trigonometry in the triangle

  • The law of sines – R\displaystyle R the radius of the circumscribed circle
    asin⁡α=bsin⁡β=csin⁡γ=2R\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R
  • The law of cosines
    c2\displaystyle c^2 =\displaystyle = a2+b2−2abcos⁡γ\displaystyle a^2+b^2-2ab\cos\gamma
  • The area of a triangle – two sides and the angle between them
    S\displaystyle S =\displaystyle = 12absin⁡γ\displaystyle \frac{1}{2}ab\sin\gamma

Vectors in the geometric approach

  • The triangle rule
    AB→+BC→\displaystyle \overrightarrow{AB}+\overrightarrow{BC} =\displaystyle = AC→\displaystyle \overrightarrow{AC}
  • A vector between two points – the end minus the start
    AB→\displaystyle \overrightarrow{AB} =\displaystyle = OB→−OA→\displaystyle \overrightarrow{OB}-\overrightarrow{OA}
  • Collinear (parallel) vectors
    v‾=t⋅u‾\displaystyle \underline{v}=t\cdot\underline{u}
  • The point M\displaystyle M divides the segment AB\displaystyle AB in the ratio m:n\displaystyle m:n
    OM→=nm+nOA→+mm+nOB→\displaystyle \overrightarrow{OM}=\frac{n}{m+n}\overrightarrow{OA}+\frac{m}{m+n}\overrightarrow{OB}AM:MB=m:n\displaystyle AM:MB=m:n
  • M\displaystyle M the midpoint of the segment AB\displaystyle AB
    OM→=12(OA→+OB→)\displaystyle \overrightarrow{OM}=\frac{1}{2}\left(\overrightarrow{OA}+\overrightarrow{OB}\right)

The scalar product

  • The length of a vector u‾=(u1,u2,u3)\displaystyle {\underline{u}}={(u_1,u_2,u_3)}
    ∣u‾∣=u‾⋅u‾=u12+u22+u32\displaystyle |\underline{u}|=\sqrt{\underline{u}\cdot\underline{u}}=\sqrt{u_1^2+u_2^2+u_3^2}
  • The scalar product in coordinates
    u‾⋅v‾\displaystyle \underline{u}\cdot\underline{v} =\displaystyle = u1v1+u2v2+u3v3\displaystyle u_1v_1+u_2v_2+u_3v_3
  • The scalar product – α\displaystyle \alpha the angle between the vectors
    u‾⋅v‾\displaystyle \underline{u}\cdot\underline{v} =\displaystyle = ∣u‾∣⋅∣v‾∣⋅cos⁡α\displaystyle |\underline{u}|\cdot|\underline{v}|\cdot\cos\alpha
  • Perpendicular vectors
    u‾⊥v‾  ⟺  u‾⋅v‾=0\displaystyle \underline{u}\perp\underline{v}\iff\underline{u}\cdot\underline{v}=0
  • A short scalar product
    (u‾+v‾)2\displaystyle (\underline{u}+\underline{v})^2 =\displaystyle = u‾2+2u‾⋅v‾+v‾2\displaystyle \underline{u}^2+2\underline{u}\cdot\underline{v}+\underline{v}^2

Points and vectors in axes

  • A vector between two points
    AB→\displaystyle \overrightarrow{AB} =\displaystyle = (xB−xA, yB−yA, zB−zA)\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)
  • The distance between two points
    AB\displaystyle AB =\displaystyle = (xB−xA)2+(yB−yA)2+(zB−zA)2\displaystyle \sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}
  • The midpoint of the segment AB\displaystyle AB
    M=(xA+xB2,yA+yB2,zA+zB2)\displaystyle M=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2},\frac{z_A+z_B}{2}\right)

The line and the plane

  • A line – v‾\displaystyle \underline{v} a point on the line, u‾\displaystyle \underline{u} the direction vector
    x‾=v‾+t⋅u‾\displaystyle \underline{x}=\underline{v}+t\cdot\underline{u}
  • A plane in parametric form – u‾\displaystyle \underline{u}, v‾\displaystyle \underline{v} not parallel
    x‾=p‾+t⋅u‾+s⋅v‾\displaystyle \underline{x}=\underline{p}+t\cdot\underline{u}+s\cdot\underline{v}
  • A plane’s equation – n‾=(a,b,c)\displaystyle {\underline{n}}={(a,b,c)} the normal vector
    ax+by+cz+d=0\displaystyle ax+by+cz+d=0
  • The plane through P(x1,y1,z1)\displaystyle P(x_1,y_1,z_1) whose normal vector is (a,b,c)\displaystyle (a,b,c)
    a(x−x1)+b(y−y1)+c(z−z1)=0\displaystyle a(x-x_1)+b(y-y_1)+c(z-z_1)=0

Distances

  • The distance of a point P(x1,y1,z1)\displaystyle P(x_1,y_1,z_1) from the plane ax+by+cz+d=0\displaystyle {ax+by+cz+d}={0}
    d\displaystyle d =\displaystyle = ∣a⋅x1+b⋅y1+c⋅z1+d∣a2+b2+c2\displaystyle \frac{|a\cdot x_1+b\cdot y_1+c\cdot z_1+d|}{\sqrt{a^2+b^2+c^2}}
  • The distance between the parallel planes ax+by+cz+d1=0\displaystyle {ax+by+cz+d_1}={0} and ax+by+cz+d2=0\displaystyle {ax+by+cz+d_2}={0}
    d\displaystyle d =\displaystyle = ∣d1−d2∣a2+b2+c2\displaystyle \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}

Angles

  • The angle between two lines – u‾\displaystyle \underline{u}, v‾\displaystyle \underline{v} the direction vectors
    cos⁡α\displaystyle \cos\alpha =\displaystyle = ∣u‾⋅v‾∣∣u‾∣⋅∣v‾∣\displaystyle \frac{|\underline{u}\cdot\underline{v}|}{|\underline{u}|\cdot|\underline{v}|}
  • The angle between the line x‾=v‾+t⋅u‾\displaystyle {\underline{x}}={\underline{v}+t\cdot\underline{u}} and the plane whose normal is n‾\displaystyle \underline{n}
    sin⁡β\displaystyle \sin\beta =\displaystyle = ∣n‾⋅u‾∣∣n‾∣⋅∣u‾∣\displaystyle \frac{|\underline{n}\cdot\underline{u}|}{|\underline{n}|\cdot|\underline{u}|}
  • The angle between two planes – n1‾\displaystyle \underline{n_1}, n2‾\displaystyle \underline{n_2} the normals
    cos⁡α\displaystyle \cos\alpha =\displaystyle = ∣n1‾⋅n2‾∣∣n1‾∣⋅∣n2‾∣\displaystyle \frac{|\underline{n_1}\cdot\underline{n_2}|}{|\underline{n_1}|\cdot|\underline{n_2}|}