Chapter 4 Geometry

4.1 Plane Geometry

4.1.1 Points

Distance between two points

\[P(x_1,y_1) \wedge Q(x_2,y_2)\]

\[\Rightarrow d(P,Q) = |PQ| = \sqrt {(x_2-x_1)^2+(y_2-y_1)^2}\]

4.1.2 Lines

Midpoint of a line segment

\[P(x_1,y_1) \wedge Q(x_2,y_2) \wedge M=\text {midpoint}_{[PQ]}\]

\[\Rightarrow co(M)=\left (\dfrac {x_1+x_2}{2},\dfrac {y_1+y_2}{2}\right )\]

Equation of a line

• • Equation of a line passing through point \(P(x_1,y_1)\) with slope \(m\):

  \[y - y_1 = m(x - x_1)\]

  or

  \[\begin {vmatrix}x & y & 1 \\ x_1 & y_1 & 1 \\ 1 & m & 0\end {vmatrix} = 0\]

  

• • Equation of a line passing through points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\):

  \[y - y_1 = \dfrac {y_2-y_1}{x_2-x_1}(x - x_1)\]

  or

  \[\begin {vmatrix}x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1\end {vmatrix} = 0\]

  

Relative position of two lines

If

\[a \leftrightarrow y = m_1 x + q_1 \wedge b \leftrightarrow y = m_2 x + q_2\]

• • then the angle between \(a\) and \(b\) is given by \(\cos (\widehat {a,b}) = \dfrac {|1+m_1m_2|}{\sqrt {1+m_1^2}\sqrt {1+m_2^2}}\)

• • \(a \parallel b\) if \(m_1=m_2\)

• • \(a \perp b\) if \(m_1m_2=-1\)

Distance from a point to a line

\[P(x_1,y_1) \wedge a \leftrightarrow ux+vy+w=0\]

\[\Rightarrow d(P,a) = \dfrac {|ux_1+vy_1+w|}{\sqrt {u^2+v^2}}\]

4.1.3 Triangles

Centroid of a triangle

\[P(x_1,y_1) \wedge Q(x_2,y_2) \wedge R(x_3,y_3)\]

\[\Rightarrow co(Z)=\left (\dfrac {x_1+x_2+x_3}{3},\dfrac {y_1+y_2+y_3}{3}\right )\]

with \(Z=\text {centroid}_{\Delta PQR}\)

4.1.4 Circles

Definition

A circle with center \(M\) and radius \(r\):

\[\mathcal {C}(M,r) = \{P \in \pi \;|\; d(P,M) = r\}\]

Equation of the circle

\[P(x,y) \in \mathcal {C}(M(a,b),r)\]

\[\Leftrightarrow (x-a)^2 + (y-b)^2 = r^2\]

4.2 Plane Figures

4.2.1 Concepts

Area: The amount of space that the figure occupies.

Perimeter: The total length of the boundary of the figure.

4.2.2 Figures

Square

• • Figure with four equal sides and four right angles (\(\adeg {90}\)).

• • \(z\) = side

• • Area:

  \[A = z^2\]

• • Perimeter:

  \[P = 4z\]

Rectangle

• • Figure with two pairs of equal sides and four right angles (\(\adeg {90}\)).

• • \(l\) = length

• • \(w\) = width

• • Area:

  \[A = l \cdot w\]

• • Perimeter:

  \[P = 2l + 2w\]

Triangle

• • Figure with three sides and three angles.

• • \(b\) = base

• • \(h\) = height

• • Area:

  \[A = \frac {1}{2} b \cdot h\]

• • Perimeter:

  \[P = a + b + c\]

  (where \(a\), \(b\), and \(c\) are the lengths of the sides)

Circle

• • Figure where all points on the boundary are equidistant from the center.

• • \(r\) = radius

• • \(d\) = diameter

• • \(O\) = center

• • Area:

  \[A = \pi r^2\]

• • Perimeter:

  \[P = 2\pi r\]

  or

  \[P = \pi d\]

Parallelogram

• • Figure with two pairs of parallel sides.

• • \(b\) = base

• • \(h\) = height

• • Area:

  \[A = b \cdot h\]

• • Perimeter:

  \[P = 2a + 2b\]

  (where \(a\) and \(b\) are the lengths of the sides)

Rhombus

• • Figure with four equal sides and opposite angles that are equal.

• • \(d_1\) = long diagonal

• • \(d_2\) = short diagonal

• • Area:

  \[A = \frac {1}{2} d_1 \cdot d_2\]

• • Perimeter:

  \[P = 4z\]

  (where \(z\) is the side length)

Trapezoid

• • Figure with one pair of parallel sides.

• • \(a\) and \(b\) = lengths of the parallel sides

• • \(h\) = height (distance between the parallel sides)

• • Area:

  \[A = \frac {1}{2} (a + b) \cdot h\]

• • Perimeter:

  \[P = a + b + c + d\]

  (where \(c\) and \(d\) are the lengths of the non-parallel sides)

Ellipse

• • Figure where the sum of the distances from any point on the boundary to two fixed points (foci) is constant.

• • \(a\) = semi-major axis (long radius)

• • \(b\) = semi-minor axis (short radius)

• • Area:

  \[A = \pi \cdot a \cdot b\]

• • Perimeter:

  \[P \approx \pi [3(a + b) - \sqrt {(3a + b)(a + 3b)}]\]

  (approximation)

Kite

• • Figure with two pairs of adjacent sides that are equal.

• • \(d_1\) = long diagonal

• • \(d_2\) = short diagonal

• • Area:

  \[A = \frac {1}{2} d_1 \cdot d_2\]

• • Perimeter:

  \[P = 2a + 2b\]

  (where \(a\) and \(b\) are the lengths of the adjacent sides)

Regular Polygon

• • Figure with equal sides and angles.

• • \(n\) = number of sides

• • \(z\) = side length

• • \(r\) = apothem (distance from the center to the midpoint of a side)

• • Area:

  \[A = \frac {1}{2} n \cdot z \cdot r\]

• • Perimeter:

  \[P = n \cdot z\]

4.3 Synthetic Solid Geometry

4.3.1 Axioms

Basic concepts

\[ \begin {array}{c|c|c} \bf symb & \bf name & \bf elements \\ \hline \Sigma & \text {space: set of points} & A,B,C,... \\ \mathcal {R} & \text {set of lines in } \Sigma & a, b, c, ... \\ \mathcal {P} & \text {set of planes in } \Sigma & \alpha , \beta , \gamma , ... \end {array} \]

Axioms

• • Space is an infinite set of points.

• • A line is an infinite, non-empty subset of space.

• • A plane is an infinite, non-empty subset of space.

• • In every plane of space, the axioms and theorems of plane geometry hold.

• • Through three non-collinear points \(A\), \(B\) and \(C\) there passes exactly one plane.

• • If two distinct points of a line belong to a plane, then this line lies in that plane.

Consequences

• • Through two distinct points there passes at least one plane.

• • Two distinct points determine exactly one line.

• • Through a line there pass infinitely many planes.

4.3.2 Relative position

Two planes \(\alpha \) and \(\beta \)

• • Coincident: \(\alpha \cap \beta = \alpha = \beta \)

• • Strictly parallel: \(\alpha \cap \beta = \emptyset \)

• • Intersecting: \(\alpha \cap \beta = \mathbf {line\ of\ intersection}\; s\)

Remark: parallel = coincident or strictly parallel

Line \(a\) and plane \(\alpha \)

• • Contained in: \(a \parallel \alpha \wedge a \cap \alpha = a \Leftrightarrow a \subset \alpha \)

• • Strictly parallel: \(a \parallel \alpha \wedge a \cap \alpha = \emptyset \)

• • Intersecting: \(a \cap \alpha = \mathbf {point\ of\ intersection}\; \{S\}\)

Remark: parallel = contained in or strictly parallel

Two lines \(a\) and \(b\)

• • Coincident: \(a \cap b = a = b\)

• • Intersecting: \(a \cap b = \mathbf {point\ of\ intersection}\; \{S\}\)

• • Strictly parallel: \(a \cap b = \emptyset \wedge a \parallel b\)

• • Skew: \(a \cap b = \emptyset \wedge a \nparallel b\)

Remark: parallel = coincident or strictly parallel

4.4 Analytic Solid Geometry

4.4.1 Equations of a line

Line through a point with a direction vector

\begin{align*} \text {line } t &\leftrightarrow \text {through } P_1(x_1,y_1,z_1) \text { with direction vector } \vec {R}(a,b,c)\\ &\leftrightarrow \vec {P} = \vec {P_1} + k \cdot \vec {R}\\ &\leftrightarrow \begin{cases} x = x_1 + k \cdot a\\ y = y_1 + k \cdot b\\ z = z_1 + k \cdot c\end {cases} \qquad (k\in \mathbb {R})\\ &\leftrightarrow \begin{pmatrix}x\\y\\z\end {pmatrix} = \begin{pmatrix}x_1\\y_1\\z_1\end {pmatrix} + k \cdot \begin{pmatrix}a\\b\\c\end {pmatrix} \qquad (k\in \mathbb {R})\\ &\leftrightarrow \dfrac {x-x_1}{a}=\dfrac {y-y_1}{b}=\dfrac {z-z_1}{c} \qquad (a,b,c \in \mathbb {R}_0) \end{align*}

Line through two points

\begin{align*} \text {line } t &\leftrightarrow \text {through } P_1(x_1,y_1,z_1) \text { and } P_2(x_2, y_2, z_2)\\ &\leftrightarrow \vec {P} = \vec {P_1} + k \cdot (\vec {P_2} - \vec {P_1}) \qquad (k\in \mathbb {R})\\ &\leftrightarrow \begin{cases} x = x_1 + k \cdot (x_2 - x_1)\\ y = y_1 + k \cdot (y_2 - y_1)\\ z = z_1 + k \cdot (z_2 - z_1)\end {cases} \qquad (k\in \mathbb {R})\\ &\leftrightarrow \begin{pmatrix}x\\y\\z\end {pmatrix} = \begin{pmatrix}x_1\\y_1\\z_1\end {pmatrix} + k \cdot \begin{pmatrix}x_2 - x_1\\y_2 - y_1\\z_2 - z_1\end {pmatrix} \qquad (k\in \mathbb {R})\\ &\leftrightarrow \dfrac {x-x_1}{x_2-x_1}=\dfrac {y-y_1}{y_2-y_1}=\dfrac {z-z_1}{z_2-z_1} \qquad (x_2 \neq x_1, y_2 \neq y_1, z_2 \neq z_1) \end{align*}

Special lines

\begin{align*} x\text {-axis} &\leftrightarrow y = z = 0\\ y\text {-axis} &\leftrightarrow x = z = 0\\ z\text {-axis} &\leftrightarrow x = y = 0 \end{align*}

4.4.2 Equations of a plane

Plane through a point with two direction vectors

\begin{align*} \text {plane } \alpha &\leftrightarrow \text {through } P_1(x_1, y_1, z_1) \text { with direction vectors } \vec {R_1}(a_1, b_1, c_1) \text { and } \vec {R_2}(a_2, b_2, c_2) \\ &\leftrightarrow \vec {P} = \vec {P_1} + k \cdot \vec {R_1} + l \cdot \vec {R_2} \qquad (k,l\in \mathbb {R}) \\ &\leftrightarrow \begin{cases} x = x_1 + k \cdot a_1 + l \cdot a_2\\ y = y_1 + k \cdot b_1 + l \cdot b_2\\ z = z_1 + k \cdot c_1 + l \cdot c_2\end {cases} \qquad (k,l\in \mathbb {R}) \\ &\leftrightarrow \begin{pmatrix}x\\y\\z\end {pmatrix} = \begin{pmatrix}x_1\\y_1\\z_1\end {pmatrix} + k \cdot \begin{pmatrix}a_1\\b_1\\c_1\end {pmatrix} + l \cdot \begin{pmatrix}a_2\\b_2\\c_2\end {pmatrix} \qquad (k,l\in \mathbb {R}) \\ &\leftrightarrow \begin{vmatrix}x-x_1 & a_1 & a_2 \\y-y_1 & b_1 & b_2 \\z-z_1 & c_1 & c_2\end {vmatrix} = 0\\ &\leftrightarrow \begin{vmatrix}x & y & z & 1 \\x_1 & y_1 & z_1 & 1 \\a_1 & b_1 & c_1 & 0 \\a_2 & b_2 & c_2 & 0\end {vmatrix}= 0 \end{align*}

Plane through three points

\begin{align*} \text {plane } \alpha &\leftrightarrow \text {through } P_1(x_1, y_1, z_1)\text {, } P_2(x_2, y_2, z_2) \text { and } P_3(x_3, y_3, z_3)\\ &\leftrightarrow \vec {P} = \vec {P_1} + k \cdot (\vec {P_2} - \vec {P_1}) + l \cdot (\vec {P_3} - \vec {P_1}) \qquad (k,l\in \mathbb {R}) \\ &\leftrightarrow \begin{cases} x = x_1 + k \cdot (x_2- x_1) + l \cdot (x_3 - x_1)\\ y = y_1 + k \cdot (y_2-y_1) + l \cdot (y_3-y_1)\\ z = z_1 + k \cdot (z_2-z_1) + l \cdot (z_3-z_1)\end {cases} \qquad (k,l\in \mathbb {R}) \\ &\leftrightarrow \begin{pmatrix}x\\y\\z\end {pmatrix} = \begin{pmatrix}x_1\\y_1\\z_1\end {pmatrix} + k \cdot \begin{pmatrix}x_2- x_1\\y_2-y_1\\z_2-z_1\end {pmatrix} + l \cdot \begin{pmatrix}x_3-x_1\\y_3-y_1\\z_3-z_1\end {pmatrix} \qquad (k,l\in \mathbb {R}) \\ &\leftrightarrow \begin{vmatrix}x & y & z & 1 \\x_1 & y_1 & z_1 & 1 \\x_2 & y_2 & z_2 & 1 \\x_3 & y_3 & z_3 & 1\end {vmatrix}= 0 \end{align*}

General form of a plane

\[\text {plane } \alpha \leftrightarrow ux + vy + wz + t = 0\]

• • \(u\), \(v\), \(w\): direction coefficients of the plane

• • \(\vec {n} = (u, v, w)\): normal vector, \(\vec {n} \perp \alpha \)

• • \(t\): constant term, position relative to \(\vec {O}\)

• • \(t = 0 \Leftrightarrow \vec {O} \in \alpha \Leftrightarrow \alpha = \alpha _O\)

Plane in intercept form

\[\dfrac {x}{a} + \dfrac {y}{b} + \dfrac {z}{c} = 1\]

• • \(a\): length of the intercept on the \(x\)-axis

• • \(b\): length of the intercept on the \(y\)-axis

• • \(c\): length of the intercept on the \(z\)-axis

• • \(a\), \(b\) and \(c\): the intercepts

Special planes

• • \(xy\)-plane: \(z=0\)

• • \(xz\)-plane: \(y=0\)

• • \(yz\)-plane: \(x=0\)

4.4.3 Relative position of lines

The lines

\begin{align*} e &\leftrightarrow \vec {P_e} = \vec {P_{e_1}} + k \cdot \vec {R_e}\\ f &\leftrightarrow \vec {P_f} = \vec {P_{f_1}} + l \cdot \vec {R_f} \end{align*} with

\begin{align*} P_{e_1}(x_1,y_1,z_1), R_e(a_1,b_1,c_1)\\ P_{f_1}(x_2,y_2,z_2), R_f(a_2,b_2,c_2) \end{align*}

Parallel

\begin{align*} e \parallel f &\Leftrightarrow \text {direction ratios are proportional}\\ &\Leftrightarrow \exists k \in \mathbb {R} : (a_1,b_1,c_1) = k \cdot (a_2, b_2, c_2) \\ &\Leftrightarrow \dfrac {a_1}{a_2} = \dfrac {b_1}{b_2} = \dfrac {c_1}{c_2} \end{align*}

Intersecting

\begin{align*} e \text { intersects } f &\Leftrightarrow \exists ! \text { solution }(k,l) : P_e = P_f\\ &\Leftrightarrow \begin{cases} (a_1,b_1,c_1) \not \sim (a_2,b_2,c_2) \\ \begin{vmatrix}a_1 & a_2 & x_2 - x_1\\b_1 & b_2 & y_2 - y_1\\c_1 & c_2 & z_2 - z_1\\\end {vmatrix} = 0 \end {cases} \end{align*}

Skew

\begin{align*} e \text { is skew to } f &\Leftrightarrow \nexists \text { solution }(k,l) : P_e = P_f\\ &\Leftrightarrow \begin{cases} (a_1,b_1,c_1) \not \sim (a_2,b_2,c_2) \\ \begin{vmatrix}a_1 & a_2 & x_2 - x_1\\b_1 & b_2 & y_2 - y_1\\c_1 & c_2 & z_2 - z_1\\\end {vmatrix} \neq 0 \end {cases} \end{align*}

4.4.4 Relative position of a line and a plane

\begin{align*} \text {line } e &\leftrightarrow \begin{cases} x = x_1 + k \cdot a\\ y = y_1 + k \cdot b\\ z = z_1 + k \cdot c\end {cases} \qquad (k\in \mathbb {R})\\ \text {plane } \alpha &\leftrightarrow ux + vy + wz + t = 0 \end{align*}

Intersecting

\[e \text { intersects } \alpha \Leftrightarrow ua + vb + wc \neq 0\]

Parallel

\[e \parallel \alpha \Leftrightarrow ua + vb + wc = 0\]

Line contained in the plane

\[e \subset \alpha \Leftrightarrow ua + vb + wc = 0 \wedge ux_1+vy_1+wz_1+t=0\]

4.4.5 Relative position of planes

\begin{align*} \text {plane } \alpha &\leftrightarrow u_1x + v_1y + w_1z + t_1 = 0\\ \text {plane } \beta &\leftrightarrow u_2x + v_2y + w_2z + t_2 = 0 \end{align*}

Intersecting

\[\alpha \text { intersects } \beta \Leftrightarrow (u_1,v_1,w_1) \not \sim (u_2,v_2,w_2)\]

• • line of intersection: \(s = \alpha \cap \beta \):

• • direction ratios of the line of intersection: \(R_s(a,b,c)\):

  \(\seteqnumber{0}{4.}{0}\)

  \begin{align*} a &= \lambda \cdot \begin{vmatrix}v_1 & w_1\\v_2 & w_2\end {vmatrix}\\ b &= - \lambda \cdot \begin{vmatrix}u_1 & w_1\\u_2 & w_2\end {vmatrix}\qquad \lambda \text { arbitrary }\in \mathbb {R}_0\\ c &= \lambda \cdot \begin{vmatrix}u_1 & v_1\\u_2 & v_2\end {vmatrix} \end{align*}

Parallel

\[\alpha \parallel \beta \Leftrightarrow (u_1,v_1,w_1) \sim (u_2,v_2,w_2)\]

Coincident

\[\alpha = \beta \Leftrightarrow (u_1,v_1,w_1,t_1) \sim (u_2,v_2,w_2,t_2)\]

4.5 Solids

4.5.1 Concepts

Volume: Amount of space occupied by the figure.

Surface area: Sum of the surfaces of all side faces.

Net: 2D representation by unfolding the side faces and laying them flat.

4.5.2 Figures

Polyhedron

3D figure with polygonal surfaces

Prism

• • Polyhedron with parallel, congruent bases connected by rectangular side faces.

• • \(B\) = base area

• • \(h\) = height

• • \(P\) = base perimeter

• • Volume:

  \[V = B \cdot h\]

• • Surface area:

  \[A = 2B + P \cdot h\]

Pyramid

• • Polyhedron with a polygonal base and triangular faces that meet at a vertex.

• • \(B\) = base area

• • \(h\) = height

• • Volume:

  \[V = \frac {1}{3} B \cdot h\]

Cube

• • Prism with six square faces, equal sides, and \(\adeg {90}\) angles.

• • \(s\) = side

• • Volume:

  \[V = s^3\]

• • Surface area:

  \[A = 6s^2\]

Rectangular Prism

• • Prism with six rectangular faces.

• • \(l\) = length

• • \(w\) = width

• • \(h\) = height

• • Volume:

  \[V = l \cdot w \cdot h\]

• • Surface area:

  \[A = 2lw + 2lh + 2wh\]

Cylinder

• • 3D figure with congruent circular bases connected by a curved surface.

• • \(r\) = radius

• • \(h\) = height

• • Volume:

  \[V = \pi r^2 h\]

• • Surface area:

  \[A = 2\pi r(r + h)\]

Cone

• • 3D figure with a circular base and a curved surface that narrows to a point.

• • \(r\) = radius

• • \(h\) = height

• • \(l\) = slant height

• • Volume:

  \[V = \frac {1}{3} \pi r^2 h\]

• • Surface area:

  \[A = \pi r(r + l)\]

Sphere

• • 3D figure with all points at equal distance from the center.

• • \(r\) = radius

• • Volume:

  \[V = \frac {4}{3} \pi r^3\]

• • Surface area:

  \[A = 4\pi r^2\]

4.6 Vectors

4.6.1 Definitions of vectors

Bound vector \(\vec {v}=\vec {AB}\) a line segment with

• • a starting point \(A\)

• • an end point \(B\)

• • a length \(|AB|\)

• • a direction, line \(AB\)

• • a sense, \(A \to B\)

Free vector a collection of line segments with

• • the same length

• • the same direction

• • the same sense

Representative one arbitrary line segment from the free vector

Position vector \(\vec {A} = \vec {OA}\) with \(O\) being the origin

Zero vector \(\vec {o}\) : length \(0\)

Notation: \(\vec {u}, \vec {v}, ...\)

Collection: \(\mathcal {V}\)

Coordinate \(co(\vec {A}) = co(A) = (x,y,z)\)

Norm of \(\vec {v} = \vec {AB}\) is the distance between \(A\) and \(B\)

\[\| \vec {v} \| = |AB|\]

4.6.2 Adding vectors

Chasles-Möbius formula

\[\vec {AB} + \vec {BC} = \vec {AC}\]

With coordinates

\(\left .\begin {aligned}co(\vec {u}) &= (x_1, y_1, z_1)\\ co(\vec {v}) &= (x_2, y_2, z_2)\end {aligned}\right \rbrace \Rightarrow \)

\[co(\vec {u} + \vec {v}) = (x_1+x_2, y_1+y_2,z_1+z_2)\]

Properties

• • I: \(\forall \vec {u}, \vec {v} \in \mathcal {V} : \vec {u} + \vec {v} \in \mathcal {V}\)

• • A: \(\forall \vec {u}, \vec {v}, \vec {w} \in \mathcal {V} : (\vec {u} + \vec {v}) + \vec {w} = \vec {u} + (\vec {v} + \vec {w})\)

• • N: \(\exists \vec {o} \in \mathcal {V}, \forall \vec {u} \in \mathcal {V} : \vec {u} + \vec {o} = \vec {u} = \vec {o} + \vec {u}\)

• • S: \(\forall \vec {u} \in \mathcal {V}, \exists ! \vec {-u} \in \mathcal {V} : \vec {u} + (\vec {-u}) = \vec {o} = (\vec {-u}) + \vec {u}\)

• • C: \(\forall \vec {u}, \vec {v} \in \mathcal {V} : \vec {u} + \vec {v} = \vec {v} + \vec {u}\)

\[\Rightarrow \mathcal {V}, + \text { is a commutative group}\]

4.6.3 Scalar multiplication of vectors

Definition

\[r \cdot \vec {v} (r\in \mathbb {R}_0, \vec {v} \in \mathcal {V}_o)\]

is a vector with

• • length equal to \(|r| \cdot \| \vec {v} \|\)

• • direction as \(\vec {v}\)

• • sense \(\begin {cases} \text {as } \vec {v} &(r>0)\\ \text {opposite } \vec {v} &(r<0)\end {cases}\)

\[0 \cdot \vec {v} = \vec {o}\]

\[r \cdot \vec {o} = \vec {o}\]

With coordinates

\(co(\vec {v}) = (x,y,z) \Rightarrow \)

\[co(r\vec {v}) = (rx,ry,rz)\]

Properties

• • \(\forall r \in \mathbb {R}, \forall \vec {v} \in \mathcal {V} : r \cdot \vec {v} \in \mathcal {V}\)

• • \(\forall r,s \in \mathbb {R}, \forall \vec {v} \in \mathcal {V} : r \cdot (s \cdot \vec {v}) = (r \cdot s) \cdot \vec {v}\)

• • \(\forall r \in \mathbb {R}, \forall \vec {u}, \vec {v} \in \mathcal {V} : r \cdot (\vec {u} + \vec {v}) = r \cdot \vec {u} + r \cdot \vec {v}\)

• • \(\forall r,s \in \mathbb {R}, \forall \vec {v} \in \mathcal {V} : (r + s) \cdot \vec {v} = r \cdot \vec {v} + s \cdot \vec {v}\)

• • \(\exists 1 \in \mathbb {R}, \forall \vec {v} \in \mathcal {V} : 1 \cdot \vec {v} = \vec {v}\)

4.6.4 Norm of a Vector

Definition

norm of \(\vec {v} = \vec {AB}\) is the distance between \(A\) and \(B\)

\[\| \vec {v} \| = d(A,B) = |AB|\]

With Coordinates

\(co(\vec {v}) = (x,y,z) \Rightarrow \)

\[\| \vec {v} \| = \sqrt {x^2 + y^2 + z^2}\]

Minkowski Inequality

\[\|\vec {u} + \vec {v}\| \leq \|\vec {u}\| + \|\vec {v}\|\]

4.6.5 Dot Product of Two Vectors

Definition

\(\vec {u}, \vec {v} \in \mathcal {V}_0\) and \(\phi \) is the angle between \(\vec {u}\) and \(\vec {v}\) \(\Rightarrow \)

\[\vec {u} \cdot \vec {v} = \|\vec {u}\| \cdot \|\vec {v}\| \cdot \cos {\phi }\]

With Coordinates

\(\left .\begin {aligned}co(\vec {u}) &= (x_1, y_1, z_1)\\ co(\vec {v}) &= (x_2, y_2, z_2)\end {aligned}\right \rbrace \Rightarrow \)

\[\vec {u} \cdot \vec {v} = x_1 \cdot x_2 + y_1 \cdot y_2 + z_1 \cdot z_2\]