12.1 Vetors in the Plane (Wk 0+1)

Denote points on a line and in the plane, geometrically and algebraically. When is each view more useful?

• Points on a line: use the real line.

  • Geometrically, a point $P$ sits at a coordinate such as $\tfrac{3}{2}$.

  • Algebraically, $\mathbb{R}=\{a : a \text{ real numbers}\}$ (bold R in the textbook). A point has a coordinate, e.g. $P=\tfrac32$.

  • $P\in\mathbb{R}$ reads “$P$ in $\mathbb{R}$”, i.e. “$P$ is an element/point in $\mathbb{R}$”.

• Points in the plane: use Cartesian (or rectangular) coordinates, i.e. the xy-plane.

  • Geometrically, a point has two coordinates, e.g. $P=(1,2)$.

  • Algebraically, $\mathbb{R}^2=\{(a,b): a,b \text{ both real numbers}\}$.

  • $P\in\mathbb{R}^2$ means “$P=(a,b)$ for some real numbers $a,b$”, e.g. $P=(1,2)\in\mathbb{R}^2$.

• Takeaway: geometry is usually more intuitive, while algebra works better for computation.

Define a vector in the plane. Name its parts, its notation, and the formula for its length.

• A vector in the plane has two points, $P$ and $Q$.

  • $P$ is the initial point / tail / basepoint.

  • $Q$ is the terminal point / head / tip.

  • It is denoted $\overrightarrow{PQ}$ or $\vec v$.

• The length (magnitude) of $\vec v$ is the distance from $P$ to $Q$, written $\|\overrightarrow{PQ}\|$.

• Formula: if $P=(a_1,b_1)$ and $Q=(a_2,b_2)$, then
  $\|\vec v\|=\|\overrightarrow{PQ}\|=\sqrt{(a_2-a_1)^2+(b_2-b_1)^2}$

• Notes’ example: $P=(-1,2)$, $Q=(2,1)$, $\vec v=\overrightarrow{PQ}$.

What is a position vector? How is it written, and what is the zero vector?

• Special case: $P=O$ (the origin), so $\vec v=\overrightarrow{OQ}$. A vector starting at $O$ is a position vector.

• For $Q=(a,b)\in\mathbb{R}^2$, write $\overrightarrow{OQ}=\langle a,b\rangle$. Example: $Q=(2,2)$.

• If $Q=O$, then $\overrightarrow{OO}=\langle 0,0\rangle$ is denoted $\vec 0$.

• Notation pitfall: the letter $O$ is for the origin, and $\vec 0$ is the number zero (as a vector).

Define parallel vectors and equivalent vectors. What is the key fact about vectors based at the origin?

• Parallel: two vectors $\vec v,\vec w$ of non-zero length are parallel if the lines through them are parallel. Parallel vectors have either the same or opposite directions.

• Equivalent: $\vec v,\vec w$ are equivalent if they have the same length and same direction. Then $\vec w$ is a translation of $\vec v$.

• ★ Key fact: every vector $\vec v$ is equivalent to a unique vector based at the origin.

How do you compute the components of $\overrightarrow{PQ}$, and how is $\overrightarrow{PQ}$ written?

• If $P=(a_1,b_1)$ and $Q=(a_2,b_2)$, then $\overrightarrow{PQ}$ is equivalent to $\overrightarrow{OR}=\langle a_2-a_1,\,b_2-b_1\rangle$.

• $a_2-a_1$ is the x-component and $b_2-b_1$ is the y-component.

• Notes’ example: $P=(-1,2)$, $Q=(1,1)$, $R=(2,-1)$, so $\overrightarrow{PQ}$ is equivalent to $\overrightarrow{OR}$.

• In most applications only length and direction matter, so components are sufficient.

• ★ Writing rule: with $a=a_2-a_1$ and $b=b_2-b_1$,
  $\overrightarrow{PQ}=\langle a,b\rangle$

Define the negative, addition and subtraction of vectors, geometrically and algebraically.

• Negative $-\vec v$:

  • Geometric: same length as $\vec v$, opposite direction.

  • Algebraic: if $\vec v=\langle a,b\rangle$, then $-\vec v=\langle -a,-b\rangle$.

• Addition $\vec v+\vec w$:

  • Geometric: move $\vec w$ to $\vec w'$ so the tail of $\vec w'$ is the head of $\vec v$. The head of $\vec v+\vec w$ is the head of $\vec w'$.

  • Example: if $\vec v=\overrightarrow{PQ}$ and $\vec w=\overrightarrow{QR}$, then $\vec v+\vec w=\overrightarrow{PR}$.

  • Algebraic: if $\vec v=\langle a,b\rangle$ and $\vec w=\langle c,d\rangle$, then $\vec v+\vec w=\langle a+c,\,b+d\rangle$.

• Subtraction:

  • Geometric: $\vec v-\vec w=\vec v+(-\vec w)$.

  • Algebraic: $\vec v-\vec w=\langle a-c,\,b-d\rangle$.

Define scalar multiplication and state the length formula and the direction rules.

• For $\vec v=\langle a,b\rangle$ and $\lambda\in\mathbb{R}$: $\lambda\vec v=\langle\lambda a,\lambda b\rangle$.

• Geometrically, $\lambda\vec v$ is parallel to $\vec v$ with length $|\lambda|\,\|\vec v\|$.

  • If $\lambda>0$: same direction as $\vec v$.

  • If $\lambda<0$: opposite direction.

  • Notes’ examples: $2\vec v$ and $-3\vec v$.

• Length formula: $\|\lambda\vec v\|=|\lambda|\,\|\vec v\|$.

State the basic properties (laws) of vector operations.

• Commutative law: $\vec v+\vec w=\vec w+\vec v$.

• Associative law: $\vec u+(\vec v+\vec w)=(\vec u+\vec v)+\vec w$.

• Distributive law for scalars: $\lambda(\vec v+\vec w)=\lambda\vec v+\lambda\vec w$.

What are $\vec i$ and $\vec j$, and how do you write any plane vector as a linear combination of them?

• $\vec i=\langle1,0\rangle$ and $\vec j=\langle0,1\rangle$.

• Any $\vec v=\langle a,b\rangle$ satisfies
  $\vec v=\langle a,0\rangle+\langle0,b\rangle=a\langle1,0\rangle+b\langle0,1\rangle=a\vec i+b\vec j$

• We say $\vec v$ can be written as a linear combination of $\vec i,\vec j$.

Define a general linear combination. When is every vector one, and why?

• A linear combination of $\vec v,\vec w$ has the form $r\vec v+s\vec w$ for some $r,s\in\mathbb{R}$.

• Claim: if $\vec v,\vec w$ are both non-zero and not parallel, then any $\vec u$ is a linear combination of $\vec v,\vec w$.

• Proof from the notes:

  1. Write $\vec v=\overrightarrow{OP}$, $\vec w=\overrightarrow{OQ}$, $\vec u=\overrightarrow{OR}$.

  2. Draw a line through $R$ parallel to $\overrightarrow{OQ}$, meeting line $OP$ at $S$.

  3. Then $\overrightarrow{OR}=\overrightarrow{OS}+\overrightarrow{SR}$, with $\overrightarrow{OS}\parallel\overrightarrow{OP}$ and $\overrightarrow{SR}\parallel\overrightarrow{OQ}$.

  4. So there are $r,s\in\mathbb{R}$ with $\overrightarrow{OS}=r\overrightarrow{OP}$ and $\overrightarrow{SR}=s\overrightarrow{OQ}$.

  5. Hence $\overrightarrow{OR}=r\overrightarrow{OP}+s\overrightarrow{OQ}$, i.e. $\vec u=r\vec v+s\vec w$.

What is a unit vector? Give its general form and the formula for the unit vector in the direction of $\vec v$.

• A unit vector has length 1. Its endpoint is on the unit circle $\{(a,b): a^2+b^2=1\}$.

• It has the form $\vec e_\theta=\langle\cos\theta,\sin\theta\rangle$.

• For any non-zero $\vec v$, $\vec e_v$ is the vector with the same direction as $\vec v$ and length 1:
  $\vec e_v=\frac{1}{\|\vec v\|}\,\vec v$

State the Triangle Inequality and its equality condition.

• Theorem: for any two vectors $\vec v,\vec w$,
  $\|\vec v+\vec w\|\le\|\vec v\|+\|\vec w\|$

• Equality holds only if $\vec v=\vec 0$, or $\vec w=\vec 0$, or $\vec w=\lambda\vec v$ for $\lambda>0$.

Prove the Triangle Inequality, and explain every equality case.

• Setup: write $\vec v=\overrightarrow{PQ}$ and $\vec w=\overrightarrow{QR}$, so $\vec v+\vec w=\overrightarrow{PR}$. The formula becomes the distance inequality $\|\overrightarrow{PR}\|\le\|\overrightarrow{PQ}\|+\|\overrightarrow{QR}\|$.

• If $P,Q,R$ form a triangle: the inequality of triangles is strict, $|PR|<|PQ|+|QR|$ (no equality).

• If $P,Q,R$ do not form a triangle, either two points are equal:

  • $P=Q$: $\vec v=\vec 0$, equality.

  • $Q=R$: $\vec w=\vec 0$, equality.

  • $P=R$: $0\le|PQ|+|QP|$, which is not an equality in general.

• Or they lie on the same line: equality iff $Q$ is between $P$ and $R$, i.e. $|PR|=|PQ|+|QR|$. Then $\vec v,\vec w$ are parallel with the same direction, so there is $\lambda>0$ with $\vec w=\lambda\vec v$.