Skip to main content

Vectors and parallelism

The triangle rule (Chasles' relation)

For any three points P,Q,RP,Q,R in the plane,

the following equality holds: P⃗Q+Q⃗R=P⃗R\vec{P}Q+\vec{Q}R=\vec{P}R

The parallelogram rule

Let u⃗,v⃗∈\vec{u},\vec{v}\in V ,P⃗Q=u⃗,\vec{P}Q=\vec{u} and P⃗T=v⃗\vec{P}T=\vec{v}

Construct the parallelogram PQRTPQRT

Then u⃗+v⃗=P⃗Q+P⃗T=P⃗Q+Q⃗R=P⃗R\vec{u}+\vec{v}=\vec{P}Q+\vec{P}T=\vec{P}Q+\vec{Q}R=\vec{P}R

Subtracting two vectors

Subtracting two vectors means adding the first vector to the opposite of the second.

u⃗−v⃗=u⃗+(−v⃗)\vec{u}-\vec{v}=\vec{u}+(-\vec{v})

Collinear vectors

Let u⃗\vec{u} be a non-zero vector and v⃗\vec{v} an arbitrary vector.

  1. If u⃗\vec{u} and v⃗\vec{v} are collinear, then there is a unique real number λ\lambda such that v⃗=λ u⃗\vec{v}=\lambda \,\vec{u}.

  2. If there exists λ∈R\lambda \in R such that v⃗=λ u⃗\vec{v}=\lambda \,\vec{u}, then u⃗\vec{u} and v⃗\vec{v} are collinear.

Let a⃗\vec{a} and b⃗\vec{b} be two non-collinear vectors. For any vector v⃗∈\vec{v}\in in V there exist α,β∈R\alpha ,\beta \in R such that v⃗=αa⃗+βb⃗\vec{v}=\alpha \vec{a}+\beta \vec{b}. The scalars α\alpha and β\beta with this property are unique.

Theorem:

  1. For any u⃗\vec{u} and v⃗∈V\vec{v}\in V, u⃗+v⃗=v⃗+u⃗\vec{u}+\vec{v}=\vec{v}+\vec{u} (commutativity)

  2. For any u⃗\vec{u}, v⃗\vec{v}, w⃗∈V\vec{w}\in V, (u⃗+v⃗)+w⃗=u⃗+(v⃗+w⃗)(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w}) (associativity)

  3. For any u⃗∈V\vec{u}\in V, u⃗+0⃗=0⃗+u⃗\vec{u}+\vec{0}=\vec{0}+\vec{u} (0⃗\vec{0} is the identity element)

  4. For any u⃗∈V\vec{u}\in V, u⃗+(−u⃗)=(−u⃗)+u⃗=0⃗\vec{u}+(-\vec{u})=(-\vec{u})+\vec{u}=\vec{0} (every vector has an opposite).

Theorem: For any α,β∈R\alpha ,\beta \in R and any u⃗\vec{u}, v⃗∈V\vec{v}\in V we have:

  1. (α+β)v⃗=αv⃗+βv⃗;(\alpha +\beta )\vec{v}=\alpha \vec{v}+\beta \vec{v};

  2. α(u⃗+v⃗)=αu⃗+αv⃗;\alpha (\vec{u}+\vec{v})=\alpha \vec{u}+\alpha \vec{v};

  3. α(βv⃗)=(αβ)v⃗;\alpha (\beta \vec{v})=(\alpha \beta )\vec{v};

  4. 1⋅v⃗=v⃗.1\cdot \vec{v}=\vec{v}.

Definition: Let OxyOxy be a rectangular coordinate system in the plane and the points A(1,0) and B(0,1). We write the vectors OA→=i⃗\overrightarrow{OA}=\vec{i}, OB→=j⃗\overrightarrow{OB}=\vec{j}; i⃗,j⃗\vec{i},\vec{j} are called the unit vectors of the coordinate axes of the system OxyOxy. The pair (i⃗,j⃗\vec{i},\vec{j}) is called the basis of the system OxyOxy.

Definition: Let OxyOxy be a rectangular coordinate system in the plane. For any point M in the plane, the vector OM→\overrightarrow{OM} is called the position vector of the point M.

Definition: In space we fix three axes OxOx, OyOy, OzOz with the same origin OO, pairwise perpendicular. The resulting system is written OxyzOxyz and is called a three-dimensional rectangular (or Cartesian) coordinate system. The orientation of the axes is usually chosen as the arrows in the adjacent figure show.

Definition: In the system OxyzOxyz, each point M(x,z,y) in space is determined by the position vector OM→=x⋅i⃗+y⋅j⃗+z⋅k⃗\overrightarrow{OM}=x\cdot \vec{i}+y\cdot \vec{j}+z\cdot \vec{k}; x,y,zx, y, z are called the coordinates of the vector OM→\overrightarrow{OM} in the basis (i⃗,j⃗,k⃗\vec{i},\vec{j},\vec{k}); x is called the abscissa, y the ordinate, and z the applicate of the point M(x,y,z).

Definition: The norm of a vector is the distance between its endpoints.

Let the points M1(x1,y1,z1){{M}_{1}}\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right) and M2(x2,y2,z2){{M}_{2}}\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right). Then ∣M1M2→∣=d(M1,M2)=(x1−x2)2+(y1−y2)2+(z1−z2)2\left| \overrightarrow{{{M}_{1}}{{M}_{2}}} \right|=d\left( {{M}_{1}},{{M}_{2}} \right)=\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}+{{\left( {{z}_{1}}-{{z}_{2}} \right)}^{2}}}.

The scalar product. For any two vectors v1→,v2→∈V3\overrightarrow{{{v}_{1}}},\overrightarrow{{{v}_{2}}}\in {{V}_{3}}, the real number v1→⋅v2→=∣v1→∣⋅∣v2→∣⋅cos⁡(α),\overrightarrow{{{v}_{1}}}\cdot \overrightarrow{{{v}_{2}}}=\left| \overrightarrow{{{v}_{1}}} \right|\cdot \left| \overrightarrow{{{v}_{2}}} \right|\cdot \cos (\alpha ), where α=m(∢(v1→,v2→))\alpha =m(\sphericalangle (\overrightarrow{{{v}_{1}}},\overrightarrow{{{v}_{2}}})) is called the scalar product of the vectors v1→ i v2→\overrightarrow{{{v}_{1}}}\text{ i }\overrightarrow{{{v}_{2}}}. If v1→ sau v2→\overrightarrow{{{v}_{1}}}\text{ sau }\overrightarrow{{{v}_{2}}} is zero, then by definition the scalar product v1→⋅v2→\overrightarrow{{{v}_{1}}}\cdot \overrightarrow{{{v}_{2}}} is zero.

i⃗⋅i⃗=1\vec{i}\cdot \vec{i}=1 j⃗⋅j⃗=1\vec{j}\cdot \vec{j}=1

k⃗⋅k⃗=1\vec{k}\cdot \vec{k}=1

i⃗⋅j⃗=0\vec{i}\cdot \vec{j}=0 i⃗⋅k⃗=0\vec{i}\cdot \vec{k}=0

j⃗⋅k⃗=0\vec{j}\cdot \vec{k}=0

Properties of the scalar product.

Let u,v,w∈V3.u, v, w\in {{V}_{3}}.

  1. u⃗⋅v⃗=v⃗⋅u⃗\vec{u}\cdot \vec{v}=\vec{v}\cdot \vec{u} (commutativity)

  2. u⃗⋅v⃗=∣u∣⋅Pr⁡uv\vec{u}\cdot \vec{v}=\left| u \right|\cdot {{\Pr }_{u}}v (Pr⁡uv{{\Pr }_{u}}v is the scalar projection of v⃗\vec{v} onto u⃗\vec{u})

  3. u⃗⋅(v⃗+w⃗)=u⃗⋅v⃗+u⃗⋅w⃗\vec{u}\cdot (\vec{v}+\vec{w})=\vec{u}\cdot \vec{v}+\vec{u}\cdot \vec{w} (distributivity)

  4. (α⋅u⃗)⋅v⃗=α(u⃗⋅v⃗)=u⃗⋅(αv⃗)(\alpha \cdot \vec{u})\cdot \vec{v}=\alpha (\vec{u}\cdot \vec{v})=\vec{u}\cdot (\alpha \vec{v}) (moving the scalar)

  5. If u⃗=x⋅i⃗+y⋅j⃗+z⋅k⃗\vec{u}=x\cdot \vec{i}+y\cdot \vec{j}+z\cdot \vec{k} and v⃗=r⋅i⃗+s⋅j⃗+t⋅k⃗\vec{v}=r\cdot \vec{i}+s\cdot \vec{j}+t\cdot \vec{k}, then u⃗⋅v⃗=x⋅r+y⋅s+z⋅t\vec{u}\cdot \vec{v}=x\cdot r+y\cdot s+z\cdot t

  6. The scalar product is zero if and only if u⃗⊥v⃗\vec{u}\bot \vec{v} or u⃗=0\vec{u}=0 or v⃗=0\vec{v}=0.

Theorem. Let u⃗=x⋅i⃗+y⋅j⃗+z⋅k⃗\vec{u}=x\cdot \vec{i}+y\cdot \vec{j}+z\cdot \vec{k} and v⃗=x1⋅i⃗+y1⋅j⃗+z1⋅k⃗\vec{v}={{x}_{1}}\cdot \vec{i}+{{y}_{1}}\cdot \vec{j}+{{z}_{1}}\cdot \vec{k} be vectors, the points M(x,y,z)M(x,y,z), M1(x1,y1,z1){{M}_{1}}({{x}_{1}},{{y}_{1}},{{z}_{1}}), and let α\alpha be the angle between the vectors u⃗\vec{u} and v⃗\vec{v}. Then cos⁡(α)=v1→⋅v2→∣v1→∣⋅∣v2→∣=x⋅x1+y⋅y1+z⋅z1x2+y2+z2⋅x12+y12+z12\cos (\alpha )=\frac{\overrightarrow{{{v}_{1}}}\cdot \overrightarrow{{{v}_{2}}}}{\left| \overrightarrow{{{v}_{1}}} \right|\cdot \left| \overrightarrow{{{v}_{2}}} \right|}=\frac{x\cdot {{x}_{1}}+y\cdot {{y}_{1}}+z\cdot {{z}_{1}}}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}\cdot \sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}}}

Definition. Let v⃗=OM→\vec{v}=\overrightarrow{OM} be the position vector of the point M. The angles which the direction (line, vector) OMOM makes with the positive directions of the coordinate axes OxOx, OyOy, OzOz are called direction angles. The cosines of these angles are called direction cosines.

Theorem. Let v⃗\vec{v} be the position vector of the point M(x,y,z)M(x,y,z). The vector v⃗\vec{v} makes with the axes the direction angles α,β,γ\alpha ,\beta ,\gamma with cos⁡(α)=xx2+y2+z2\cos (\alpha )=\frac{x}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}, cos⁡(β)=yx2+y2+z2\cos (\beta )=\frac{y}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}, cos⁡(γ)=zx2+y2+z2\cos (\gamma )=\frac{z}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}.

cos⁡2(α)+cos⁡2(β)+cos⁡2(γ)=1{{\cos }^{2}}(\alpha )+{{\cos }^{2}}(\beta )+{{\cos }^{2}}(\gamma )=1

If cos⁡(α),cos⁡(β),cos⁡(γ)\cos (\alpha ), \cos (\beta ), \cos (\gamma ) are the direction cosines of the line d and p≥0p\ge 0, then x⋅cos⁡(α)+y⋅cos⁡(β)+z⋅cos⁡(γ)−p=0x\cdot \cos (\alpha )+y\cdot \cos (\beta )+z\cdot \cos (\gamma )-p=0 is the normal equation of the plane perpendicular to the line d, at distance p from the origin.

The equation of the line passing through the point M(xM,yM)M\left( {{x}_{M}},{{y}_{M}} \right) with direction vector u⃗(α,β)\vec{u}\left( \alpha ,\beta \right) is: y−yM=βα(x−xM)y-{{y}_{M}}=\frac{\beta }{\alpha }\left( x-{{x}_{M}} \right).

The equation of the line passing through the points A(xA,yA)A\left( {{x}_{A}},{{y}_{A}} \right), B(xB,yB)B\left( {{x}_{B}},{{y}_{B}} \right) is: y−yA=yB−yAxB−xA(x−xA)y-{{y}_{A}}=\frac{{{y}_{B}}-{{y}_{A}}}{{{x}_{B}}-{{x}_{A}}}\left( x-{{x}_{A}} \right).

The distinct lines y=mx+ny=mx+n and y=m′x+n′y=m'x+n' are parallel if and only if m=m′m=m', and perpendicular if and only if mm′=−1mm'=-1.