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Formulas: space geometry and vectors, grade 12 (4 units)

The right-angled triangle and a triangle's area

  • Pythagoras’ theorem – c\displaystyle c the hypotenuse
    a2+b2\displaystyle a^2+b^2 =\displaystyle = c2\displaystyle c^2
  • a\displaystyle a the leg opposite the angle α\displaystyle \alpha, c\displaystyle c the hypotenuse
    sin⁡α\displaystyle \sin\alpha =\displaystyle = ac\displaystyle \frac{a}{c}
  • b\displaystyle b the leg next to the angle α\displaystyle \alpha
    cos⁡α\displaystyle \cos\alpha =\displaystyle = bc\displaystyle \frac{b}{c}
  • The leg opposite the angle over the leg next to it
    tan⁡α\displaystyle \tan\alpha =\displaystyle = ab\displaystyle \frac{a}{b}
  • The cosine of an obtuse angle
    cos⁡(180∘−α)\displaystyle \cos(180^{\circ}-\alpha) =\displaystyle = −cos⁡α\displaystyle -\cos\alpha
  • A triangle’s area – two sides and the angle between them
    S\displaystyle S =\displaystyle = 12absin⁡γ\displaystyle \frac{1}{2}ab\sin\gamma

The prism

  • The surface area – the lateral surface and the two bases; B\displaystyle B the base’s area
    S\displaystyle S =\displaystyle = M+2B\displaystyle M+2B
  • The lateral area of a right prism – P\displaystyle P the base’s perimeter, h\displaystyle h the height
    M\displaystyle M =\displaystyle = P⋅h\displaystyle P\cdot h
  • The volume of a prism – h\displaystyle h the height, the distance between the bases
    V\displaystyle V =\displaystyle = B⋅h\displaystyle B\cdot h

The pyramid

  • The surface area – the lateral surface and the base
    S\displaystyle S =\displaystyle = M+B\displaystyle M+B
  • The volume of a pyramid – a third of the prism with the same base and height
    V\displaystyle V =\displaystyle = B⋅h3\displaystyle \frac{B\cdot h}{3}

Vectors, geometrically

  • The triangle rule
    AB→+BC→\displaystyle \overrightarrow{AB}+\overrightarrow{BC} =\displaystyle = AC→\displaystyle \overrightarrow{AC}
  • The difference of two vectors leaving the same point
    AD→−AB→\displaystyle \overrightarrow{AD}-\overrightarrow{AB} =\displaystyle = BD→\displaystyle \overrightarrow{BD}
  • Collinear vectors
    v⃗=t⋅u⃗\displaystyle \vec{v}=t\cdot\vec{u}
  • A linear combination of u⃗\displaystyle \vec{u} and v⃗\displaystyle \vec{v}
    w⃗=su⃗+tv⃗\displaystyle \vec{w}=s\vec{u}+t\vec{v}

Vectors in coordinates

  • The vector between two points – the end minus the start
    AB→\displaystyle \overrightarrow{AB} =\displaystyle = (xB−xA, yB−yA, zB−zA)\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)
  • A vector’s length
    ∣u⃗∣\displaystyle |\vec{u}| =\displaystyle = x2+y2+z2\displaystyle \sqrt{x^2+y^2+z^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 scalar product

  • The definition – α\displaystyle \alpha the angle between the vectors
    u⃗⋅v⃗\displaystyle \vec{u}\cdot\vec{v} =\displaystyle = ∣u⃗∣⋅∣v⃗∣⋅cos⁡α\displaystyle |\vec{u}|\cdot|\vec{v}|\cdot\cos\alpha
  • In coordinates
    u⃗⋅v⃗\displaystyle \vec{u}\cdot\vec{v} =\displaystyle = x1x2+y1y2+z1z2\displaystyle x_1x_2+y_1y_2+z_1z_2
  • The angle between two vectors
    cos⁡α\displaystyle \cos\alpha =\displaystyle = u⃗⋅v⃗∣u⃗∣⋅∣v⃗∣\displaystyle \frac{\vec{u}\cdot\vec{v}}{|\vec{u}|\cdot|\vec{v}|}
  • A vector’s length by the scalar product
    ∣u⃗∣\displaystyle |\vec{u}| =\displaystyle = u⃗2\displaystyle \sqrt{\vec{u}^2}
  • A short scalar product
    (u⃗+v⃗)2\displaystyle (\vec{u}+\vec{v})^2 =\displaystyle = u⃗2+2u⃗⋅v⃗+v⃗2\displaystyle \vec{u}^2+2\vec{u}\cdot\vec{v}+\vec{v}^2
  • A short scalar product
    (u⃗+v⃗)⋅(u⃗−v⃗)\displaystyle (\vec{u}+\vec{v})\cdot(\vec{u}-\vec{v}) =\displaystyle = u⃗2−v⃗2\displaystyle \vec{u}^2-\vec{v}^2
  • Perpendicular vectors
    u⃗⊥v⃗  ⟺  u⃗⋅v⃗=0\displaystyle \vec{u}\perp\vec{v}\iff\vec{u}\cdot\vec{v}=0