Vectors

Vectors have magnitude and direction.

Example: Speed has a value and a direction.

Definition

The vector $\overrightarrow{PQ}$ is the directed line segment with initial point P and terminal point Q.

Conventions and notation

• Two vectors with the same length and the same direction are considered to be the same.
• The name of a vector can be written in bold: v or underlined v

Standard position and component form

Defintion

• A vector is in standard position if the origin is its initial point.
• If the vector equals OP where O is the origin and P = (x, y, z), then we write v in component form: $v=⟨x,y,z⟩$

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If vector v has initial point $P⟨x_1,x_2,x_3⟩$ and terminal point $Q⟨x_2,y_2,y_3⟩$ then

$v=\overrightarrow{PQ}=⟨x_2-x_1,y_2-y_1,z_2-z_1⟩$

The length of a vector

Definition

The length of the vector v with initial point P and terminal point Q is denoted as $|v|$ and is defined as the distance between P and Q.

• $|v|=|PQ|$

• If $v=⟨x,y,z⟩$, then $|v|=\sqrt{x^2+y^2+z^2}$ (in standard position)

• If $v=\overrightarrow{PQ}$ then $|v|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Unit Vectors

Definition

A unit vector is a vector with length = 1.

• to get a unit vector from v: $v\ne 0$, then $\frac{v}{|v|}$ is a unit vector

Direction

Definition

The direction of a non-zero vector v is the vector $\frac{v}{|v|}$

• Directions are unit vectors.
• If a direction is in standard position, its end point lies on the unit circle (in ${\mathbb{R}}^2$), or the unit sphere (in ${\mathbb{R}}^3$)

Special vectors

• zero vector: all components are 0, notaion: 0
• standard base vectors:
  • ${\mathbb{R}}^2⟨1,0⟩⟨0,1⟩$
  • ${\mathbb{R}}^3\text{:}⟨1,0,0⟩⟨0,1,0⟩⟨0,0,1⟩$

Scalar multiplication of vectors

Scalar multiplication of a vector v with a real number a results in a shorter or longer vector with the same or opposite direction as v.

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If $v=⟨v_1,...,v_n⟩$

Then $av=⟨a{v}_1,...,a{v}_n⟩\mathrm{\forall }a\in \mathbb{R}$

Sum of two vectors

Two visual ways:

• head to tail construction: shift v such that the initial point of v is the same as the terminal point of u.

• parallelogram construction: shift v such that the initial points if u and v coincide.

algebraically:

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If $u=⟨u_1,...,u_n⟩v=⟨v_1,...,v_n⟩$

Then $u+v=⟨u_1+v_1,...u_n+v_n⟩$

Properties

• $u+v=v+u$
• $(u+v)+w=u+(v+w)$
• $u+0=u$
• $u+(-u)=0$
• $0u=0$
• $1u=u$
• $a(b\mathit{u})=(ab)\mathit{u}$
• a(u + v) = au + av
• (a + b)u = au + bu

The difference of two vectors.

visually:

• head to head: shift the vectors such that the initial points coincide
• v - u is the vector from the head of u to the head of v.
• u - v is the vector from the head of v to the head of u.

Dot product

Definition

The dot product of two vectors v and u:

$u\cdot v=v_1{u}_1+v_2{u}_2+...+u_n{u}_v$

• the dot product is a number.
• the dot product can be used to describe the length of the vector: $|v|=\sqrt{v\cdot v}$

Properties

• $u\cdot v=v\cdot u$
• $(cu)\cdot v=u\cdot (cv)=c(u\cdot v)$
• $u\cdot (v+w)=u\cdot v+u\cdot w$
• $u\cdot u=|u{|}^2$
• $u\cdot 0=0$

The angle between unit vectors.

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if u and v are unit vectors, and $\theta$ is the angle between u and v, then:

$u\cdot v=\cos \theta$

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if u and v are non-zero vectors and $\theta$ is the angle between u and v, then

$u\cdot v=|u||v|\cos \theta$

Orthogonality

Definition

two vectors u and v are orthogonal or perpendicular if $u\cdot v=0$

• if $u\ne 0,v\ne 0$, then the angle between u and v is $\frac{\pi }{2}$
• if u = 0 or v = 0 then there is no angle between u and v but we still say that u and v are orthogonal.
• we denote orthogonal vectors with the "$\perp$" symbol : $v\perp u$

Projection

• $\overrightarrow{w}-c\overrightarrow{v}\perp \overrightarrow{v}$ in other words: $(\overrightarrow{w}-a\overrightarrow{v})\cdot \overrightarrow{v}=0$

Definition

Let w and v be two vectors, $v\ne 0$ The Projection of w onto v is: $pro{j}_v(w)=(\frac{w\cdot v}{|v{|}^2})v$

Cross Product

• Only in 3 dimensions => vector has 3 components

Definition

Let $u=⟨u_1,u_2,u_3⟩,v=⟨v_1,v_2,v_3⟩$ be two vectors in ${\mathbb{R}}^3$ the cross product of u and b is:

$u\times v=⟨u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1⟩$

Cross Product Template:

1. Write u and v in a column:
   • $u=⟨u_1,u_2,u_3⟩$
   • $v=⟨v_1,v_2,v_3⟩$
2. copy the first two leftmost entries of u and v.
   • $u=⟨u_1,u_2,u_3⟩u_1{u}_2$
   • $v=⟨v_1,v_2,v_3⟩v_1{v}_2$
3. calculate the first entry of u x v, starting with u_2 in an x shape.
   • $u_2{v}_3-u_3{v}_2$
4. calculate the second entry of u x v.
   • $u_3{v}_1-u_1{v}_3$
5. calculate the third entry of u x v.
   • $u_1{v}_2-u_2{v}_1$

Properties:

• $(r\overrightarrow{u})\times (s\overrightarrow{v})=(rs)(\overrightarrow{u}\times \overrightarrow{v})$
• $u\times (v+w)=u\times v+u\times w$
• $u\times v=-(v\times u)$
• $(v+w)\times u=v\times u+w\times u$
• $0\times u=u\times 0=0$
• $u\times (v\times w)\ne (u\times v)\times w$

Geometric Properties:

• $u\times v\perp uandu\times v\perp v$
• $|u\times v|=|u||v|\sin \theta$ with $\theta$ being the angle between u an v.
• right hand rule for $u\times v$:
  • index finger: first vector (u)
  • middle finger: second vector (v)
  • thumb: cross product ($u\times v$)

Lines and Planes

Definition

The line through P parallel to v is given by $\{p+tv\in {\mathbb{R}}^3|t\in \mathbb{R}\}$

where $p=\overrightarrow{OP}$

• The vector p is called a support vector of l.
• the vector v is called a direction vector of l.

Planes in ${\mathbb{R}}^3$

• A normal vector of M is a non-zero vector orthogonal to M.
• A plane is determined by:
  • three points, not on one line.
  • by a support vector and a normal vector.

Planes by normal vector:

• If n is a normal vector of M, then for every $X\in Mn\perp \overrightarrow{PX}$
• if $p=\overrightarrow{OP}\text{ and }x=\overrightarrow{OX}\text{ , then }n\cdot (x-p)=0$ this equation is called the normal equation of M.

Planes determined by three points:

How to find a normal vector: If P, Q and T are three points (not on a line), a normal vector of the plane can be found by taking a cross product, for example: $n=\overrightarrow{PQ}\times \overrightarrow{PR}$

Method: Finding a plane equation from three points

The Plane W goes through three points A, B and C.

Determine an equation for the plane W of the form $ax+by+cz-d=0$

1. Find a normal vector for the plane W.
   • Find a vector: $\overrightarrow{AB}=⟨B_1-A_1,B_2-A_2,B_3-A_3⟩$
   • Find another vector: $\overrightarrow{AC}=⟨C_1-A_1,C_2-A_2,C_3-A_3⟩$
   • Find the cross product of the vectors AB and AC (we use u and v for readability) $n=u\times v=⟨u_2{v}_3-u_3{v}_2,u_3{v}_1-u_1{v}_3,u_1{v}_2-u_2{v}_1⟩$
2. Find a support vector p, which is a vector from the origin to any point on the plane.
   • Choose $p=\overrightarrow{OA}$
3. Replace n and p in $n\cdot (x-p)=0$ (x can be replaced with $⟨x,y,z⟩$)