Comprehensive Notes on Vectors, Products, Lines & Planes

Scalars and Vectors

A scalar is a quantity that possesses magnitude only, whereas a vector possesses both magnitude and a specific direction. Geometrically, a vector is represented by a directed line segment (an arrow) whose length equals the magnitude and whose arrow–head indicates direction. A vector whose initial point is $A$ and terminal point $B$ is denoted $\overrightarrow{AB}$ and its magnitude by $\lVert \overrightarrow{AB} \rVert$.

Two vectors $\mathbf v$ and $\mathbf w$ are considered equal ($\mathbf v = \mathbf w$) if they have identical magnitudes and point in exactly the same direction; i.e. their graphical representatives are parallel, of the same length, and oriented identically.

Vector Operations in the Plane

Addition (Parallelogram / Triangle Rule)

If a particle is displaced successively by $\overrightarrow{AB}$ then $\overrightarrow{BC}$, the net displacement is $\overrightarrow{AC}$. Hence
$\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}.$ Graphically, placing the tail of one vector at the head of the other and drawing the diagonal of the resulting parallelogram produces the same resultant.

Scalar Multiplication

For a scalar $c\neq0$ and a vector $\mathbf v$, the product $c\mathbf v$ is a vector collinear with $\mathbf v$ whose length is $|c|\,\lVert \mathbf v \rVert$. Two non-zero vectors are parallel iff one is a non-zero scalar multiple of the other.

Difference of Two Vectors

The difference $\mathbf v-\mathbf w$ is defined by
$\mathbf v-\mathbf w = \mathbf v + (-\mathbf w).$
Graphically, translate $\mathbf w$ so its tail coincides with the tail of $\mathbf v$, reverse it to obtain $-\mathbf w$, then add using the parallelogram rule.

Analytical Representation in $\mathbb R^{2}$

If $P(v<em>1,v</em>2)$ is the terminal point of a vector whose tail is at the origin, the
position vector is $\langle v<em>1,v</em>2\rangle.$

Definition.
A (plane) vector is the ordered pair $\mathbf v=\langle v<em>1,v</em>2\rangle.$

Zero vector : $\mathbf0 = \langle0,0\rangle.$

Given $P<em>1(x</em>1,y<em>1)$ and $P</em>2(x<em>2,y</em>2)$,
$\overrightarrow{P<em>1P</em>2}=\langle x<em>2-x</em>1,\;y<em>2-y</em>1 \rangle.$

Magnitude

$\lVert\mathbf v\rVert = \sqrt{v<em>1^{2}+v</em>2^{2}}.$

Component-wise Operations

For $\mathbf u=\langle u<em>1,u</em>2\rangle$ and $\mathbf v=\langle v<em>1,v</em>2\rangle$:
$\mathbf u+\mathbf v = \langle u<em>1+v</em>1,\;u<em>2+v</em>2\rangle,$
$c\mathbf u = \langle cu<em>1,\;cu</em>2\rangle.$

Illustrative Examples

1. $A(-3,-2)$, $B(2,1):$ $\overrightarrow{AB}=\langle5,3\rangle.$

2. Vectors $\overrightarrow{AB}$ with $B(3,2)$ and $\overrightarrow{CD}$ with $D(4,5)$ share length $\sqrt{13}$ and slope $\tfrac23$, hence they are equal.

3. Add $\langle3,-2\rangle+\langle-1,3\rangle = \langle2,1\rangle$.

4. Two ships begin together: $\mathbf A = 15\mathbf i$, $\mathbf B = 30(\cos\tfrac\pi4\,\mathbf i+\sin\tfrac\pi4\,\mathbf j)$. Relative velocity $\mathbf D=\mathbf B-\mathbf A=15(\sqrt2-1)\mathbf i+15\sqrt2\,\mathbf j$ has magnitude $22.1\text{ km/h}.$

5. Force–balance on a sky-diver: $\mathbf g=\langle0,-180\rangle$, $\mathbf r=\langle30,180\rangle\Rightarrow\mathbf g+\mathbf r=\langle30,0\rangle$ (purely horizontal).

6. Heading problem: with wind $\mathbf w=\langle20,30\rangle$ and air-speed $400$-mph, plane’s velocity must be $\mathbf v=\langle-159.1,-30\rangle$ producing a track due west.

7. Two-rope suspension: tensions $\mathbf F<em>1=50(\mathbf i+\mathbf j),\;\mathbf F</em>2=50(-\mathbf i+\mathbf j)$ each of magnitude $50\sqrt2$ N.

Unit Vectors and Standard Basis ($\mathbf i,\mathbf j$)

Any non-zero vector $\mathbf w$ has an associated unit vector $\mathbf u=\frac{\mathbf w}{\lVert\mathbf w\rVert}.$ In the plane, $\mathbf i=\langle1,0\rangle,\;\mathbf j=\langle0,1\rangle$ form the standard basis so that $\mathbf u= u<em>1\mathbf i+u</em>2\mathbf j.$

Fundamental Properties (Axioms)

For vectors $\mathbf u,\mathbf v,\mathbf w$ and scalars $c,d$:

1. $\mathbf u+\mathbf v = \mathbf v+\mathbf u$ (commutativity)

2. $(\mathbf u+\mathbf v)+\mathbf w = \mathbf u+(\mathbf v+\mathbf w)$ (associativity)

3. $\mathbf u+\mathbf0 = \mathbf u$ ; $\mathbf u+(-\mathbf u)=\mathbf0$

4. $c(\mathbf u+\mathbf v)=c\mathbf u+c\mathbf v$

5. $c(d\mathbf u)=(cd)\mathbf u$

6. $(c+d)\mathbf u=c\mathbf u+d\mathbf u$

7. $1\mathbf u=\mathbf u$

Three-Dimensional Vectors

A vector in $\mathbb R^{3}$ is $\mathbf a=\langle a<em>1,a</em>2,a<em>3\rangle.$ Magnitude $\lVert\mathbf a\rVert=\sqrt{a</em>1^{2}+a<em>2^{2}+a</em>3^{2}}.$

Standard basis: $\mathbf i=\langle1,0,0\rangle,\;\mathbf j=\langle0,1,0\rangle,\;\mathbf k=\langle0,0,1\rangle.$
Thus $\mathbf a=a<em>1\mathbf i+a</em>2\mathbf j+a_3\mathbf k.$

Given points $P<em>1(x</em>1,y<em>1,z</em>1)$, $P<em>2(x</em>2,y<em>2,z</em>2)$
$\overrightarrow{P<em>1P</em>2}=\langle x<em>2-x</em>1,\;y<em>2-y</em>1,\;z<em>2-z</em>1\rangle.$

Example.
$P(2,-1,2)$, $Q(1,4,5)$ give $\overrightarrow{PQ}=\langle-1,5,3\rangle,$ length $\sqrt{35}$; unit vector $\left\langle\tfrac{-1}{\sqrt{35}},\tfrac{5}{\sqrt{35}},\tfrac{3}{\sqrt{35}}\right\rangle.$

Dot (Scalar) Product

For $\mathbf a=\langle a<em>1,a</em>2,a<em>3\rangle$, $\mathbf b=\langle b</em>1,b<em>2,b</em>3\rangle$:
$\mathbf a\cdot\mathbf b = a<em>1b</em>1+a<em>2b</em>2+a<em>3b</em>3.$

Properties mirror those of ordinary multiplication (commutative, distributive, scalar associative). One crucial identity:
$\mathbf a\cdot\mathbf a = \lVert\mathbf a\rVert^{2}.$

Angle Between Two Vectors

For non-zero $\mathbf a,\mathbf b$ and angle $\theta$ between them,
$\cos\theta = \frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf a\rVert\,\lVert\mathbf b\rVert}.$

Orthogonality: $\mathbf a\perp\mathbf b\iff \mathbf a\cdot\mathbf b=0$.

Inequalities

Cauchy–Schwarz: $|\mathbf a\cdot\mathbf b|\le\lVert\mathbf a\rVert\,\lVert\mathbf b\rVert.$
Triangle: $\lVert\mathbf a+\mathbf b\rVert\le\lVert\mathbf a\rVert+\lVert\mathbf b\rVert.$

Direction Cosines

For non-zero $\mathbf a$ let $\alpha,\beta,\gamma$ be angles with the positive axes; then
$\cos\alpha=\frac{a<em>1}{\lVert\mathbf a\rVert},\;\cos\beta=\frac{a</em>2}{\lVert\mathbf a\rVert},\;\cos\gamma=\frac{a_3}{\lVert\mathbf a\rVert},$
with $\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1.$

Projection and Component

Scalar component of $\mathbf b$ along $\mathbf a$:
$\operatorname{comp}_{\mathbf a}\mathbf b = \frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf a\rVert}.$

Vector projection:
$\operatorname{proj}<em>{\mathbf a}\mathbf b = \frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf a\rVert^{2}}\,\mathbf a.$ Any vector decomposes as $\mathbf b=\operatorname{proj}</em>{\mathbf a}\mathbf b+\big(\mathbf b-\operatorname{proj}_{\mathbf a}\mathbf b\big).$

Work

If a constant force $\mathbf F$ moves an object by displacement $\mathbf d$, the work is
$W=\mathbf F\cdot\mathbf d.$

Cross (Vector) Product

For $\mathbf a=a<em>1\mathbf i+a</em>2\mathbf j+a<em>3\mathbf k$ and $\mathbf b=b</em>1\mathbf i+b<em>2\mathbf j+b</em>3\mathbf k$:
\mathbf a\times\mathbf b=
\begin{vmatrix}
\mathbf i & \mathbf j & \mathbf k\
a1 & a2 & a3\ b1 & b2 & b3
\end{vmatrix}
=(a2b3-a3b2)\mathbf i+(a3b1-a1b3)\mathbf j+(a1b2-a2b1)\mathbf k.

Key facts:

1. $\mathbf a\times\mathbf b$ is orthogonal to both $\mathbf a$ and $\mathbf b$ (right-hand rule gives orientation).

2. Magnitude $\lVert\mathbf a\times\mathbf b\rVert=\lVert\mathbf a\rVert\,\lVert\mathbf b\rVert\sin\theta.$

3. $\mathbf a\times\mathbf a=\mathbf0$; vectors are parallel iff cross product vanishes.

Area and Volume

Area of parallelogram spanned by $\mathbf a,\mathbf b$ equals $\lVert\mathbf a\times\mathbf b\rVert$. Triangle area is half of that.

Scalar triple product \mathbf a\cdot(\mathbf b\times\mathbf c)=\begin{vmatrix}a1&a2&a3\b1&b2&b3\c1&c2&c_3\end{vmatrix} equals (signed) volume of the parallelepiped defined by $\mathbf a,\mathbf b,\mathbf c$. Zero indicates coplanarity.

Torque

For force $\mathbf F$ applied at position $\mathbf r$, torque $\boldsymbol\tau=\mathbf r\times\mathbf F,$ magnitude $\lVert\boldsymbol\tau\rVert=\lVert\mathbf r\rVert\,\lVert\mathbf F\rVert\sin\theta.$

Lines in Space

Given point $P<em>0(x</em>0,y<em>0,z</em>0)$ and direction vector $\mathbf v=\langle a,b,c\rangle$.

Parametric form:
$x=x<em>0+at,\;y=y</em>0+bt,\;z=z_0+ct.$

Symmetric form (when $a,b,c\neq0$):
$\frac{x-x<em>0}{a}=\frac{y-y</em>0}{b}=\frac{z-z_0}{c}.$

Two lines are
• Parallel if direction vectors are parallel.
• Intersecting when some $t<em>1,t</em>2$ satisfy both sets of parametric equations.
• Skew when neither parallel nor intersecting.

Distance from point $Q$ to line through $P$ with direction $\mathbf v$:
$D=\frac{\lVert \overrightarrow{PQ}\times\mathbf v\rVert}{\lVert\mathbf v\rVert}.$

Distance between parallel lines: choose $P$ on one, $Q$ on the other and use same formula.

Planes in Space

A plane through $P<em>0(x</em>0,y<em>0,z</em>0)$ with normal $\mathbf n=\langle a,b,c\rangle$ has equation
$a(x-x<em>0)+b(y-y</em>0)+c(z-z_0)=0.$

If two planes have normals $\mathbf n<em>1,\mathbf n</em>2$:
• They are parallel when $\mathbf n<em>1\parallel\mathbf n</em>2$.
• They are orthogonal when $\mathbf n<em>1\cdot\mathbf n</em>2=0$.

Angle $\theta$ between planes equals the angle between normals:
$\cos\theta=\frac{\mathbf n<em>1\cdot\mathbf n</em>2}{\lVert\mathbf n<em>1\rVert\,\lVert\mathbf n</em>2\rVert}.$

Line of intersection of two non-parallel planes is obtained by solving their simultaneous equations or by taking direction $\mathbf n<em>1\times\mathbf n</em>2$.

Distance from point $P<em>1$ to plane with normal $\mathbf n$ through $P</em>0$:
$D=\frac{|\mathbf n\cdot\overrightarrow{P<em>1P</em>0}|}{\lVert\mathbf n\rVert}.$

Representative Worked Problems

• Through $P(-3,3,-2)$ and $Q(2,-1,4)$, the direction vector $\overrightarrow{PQ}=\langle5,-4,6\rangle$ gives the line $x=-3+5t,\;y=3-4t,\;z=-2+6t.$ It intersects the $xy$-plane at $t=\tfrac13\Rightarrow(-\tfrac43,\,\tfrac53,0)$.

• Plane through $P(1,3,2),Q(3,-1,6),R(5,2,0)$: normals from $\overrightarrow{PQ}\times\overrightarrow{PR}$ give $6x+10y+7z=50.$

• Distance from $(1,-2,3)$ to line $x=3+t,y=-1-2t,z=4+t$ equals $\frac{\sqrt{35}}{\sqrt6}.$

• Volume of parallelepiped with edges $\mathbf a=\mathbf i+2\mathbf j+3\mathbf k$, $\mathbf b=4\mathbf i+5\mathbf j+6\mathbf k$, $\mathbf c=7\mathbf i+8\mathbf j$ is $|\mathbf a\cdot(\mathbf b\times\mathbf c)|=27.$

Summary of Cross & Dot Interaction Identities

1. $\mathbf a\times\mathbf b = -\mathbf b\times\mathbf a$ (anti-commutative)

2. $\mathbf a\times(\mathbf b+\mathbf c)=\mathbf a\times\mathbf b+\mathbf a\times\mathbf c$

3. $(\mathbf a+\mathbf b)\times\mathbf c=\mathbf a\times\mathbf c+\mathbf b\times\mathbf c$

4. $c(\mathbf a\times\mathbf b)=(c\mathbf a)\times\mathbf b=\mathbf a\times(c\mathbf b)$

5. $\mathbf a\cdot(\mathbf b\times\mathbf c)=(\mathbf a\times\mathbf b)\cdot\mathbf c$ (scalar triple cyclicity)

6. $\mathbf a\times(\mathbf b\times\mathbf c)= (\mathbf a\cdot\mathbf c)\,\mathbf b-(\mathbf a\cdot\mathbf b)\,\mathbf c.$

These results underpin numerous physical and geometric computations from torque and work to areas, volumes, and distances in analytic geometry.