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 AB→\overrightarrow{AB} and its magnitude by ∥AB→∥\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 AB→\overrightarrow{AB} then BC→\overrightarrow{BC}, the net displacement is AC→\overrightarrow{AC}. Hence
AC→=AB→+BC→.\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∣ ∥v∥|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
v−w=v+(−w).\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 $P(v1,v2)$ is the terminal point of a vector whose tail is at the origin, the
position vector is ⟨v<em>1,v</em>2⟩.\langle v<em>1,v</em>2\rangle.

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

Zero vector : 0=⟨0,0⟩.\mathbf0 = \langle0,0\rangle.

Given $P1(x1,y1)$ and $P2(x2,y2)$,
P<em>1P</em>2→=⟨x<em>2−x</em>1,  y<em>2−y</em>1⟩.\overrightarrow{P<em>1P</em>2}=\langle x<em>2-x</em>1,\;y<em>2-y</em>1 \rangle.

Magnitude

∥v∥=v<em>12+v</em>22.\lVert\mathbf v\rVert = \sqrt{v<em>1^{2}+v</em>2^{2}}.

Component-wise Operations

For $\mathbf u=\langle u1,u2\rangle$ and $\mathbf v=\langle v1,v2\rangle$:
u+v=⟨u<em>1+v</em>1,  u<em>2+v</em>2⟩,\mathbf u+\mathbf v = \langle u<em>1+v</em>1,\;u<em>2+v</em>2\rangle,
cu=⟨cu<em>1,  cu</em>2⟩.c\mathbf u = \langle cu<em>1,\;cu</em>2\rangle.

Illustrative Examples

  1. $A(-3,-2)$, $B(2,1):$ AB→=⟨5,3⟩.\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 D=B−A=15(2−1)i+152 j\mathbf D=\mathbf B-\mathbf A=15(\sqrt2-1)\mathbf i+15\sqrt2\,\mathbf j has magnitude 22.1 km/h.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 v=⟨−159.1,−30⟩\mathbf v=\langle-159.1,-30\rangle producing a track due west.

  7. Two-rope suspension: tensions F<em>1=50(i+j),  F</em>2=50(−i+j)\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 u=w∥w∥.\mathbf u=\frac{\mathbf w}{\lVert\mathbf w\rVert}. In the plane, i=⟨1,0⟩,  j=⟨0,1⟩\mathbf i=\langle1,0\rangle,\;\mathbf j=\langle0,1\rangle form the standard basis so that u=u<em>1i+u</em>2j.\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 a=⟨a<em>1,a</em>2,a<em>3⟩.\mathbf a=\langle a<em>1,a</em>2,a<em>3\rangle. Magnitude ∥a∥=a</em>12+a<em>22+a</em>32.\lVert\mathbf a\rVert=\sqrt{a</em>1^{2}+a<em>2^{2}+a</em>3^{2}}.

Standard basis: i=⟨1,0,0⟩,  j=⟨0,1,0⟩,  k=⟨0,0,1⟩.\mathbf i=\langle1,0,0\rangle,\;\mathbf j=\langle0,1,0\rangle,\;\mathbf k=\langle0,0,1\rangle.
Thus a=a<em>1i+a</em>2j+a3k.\mathbf a=a<em>1\mathbf i+a</em>2\mathbf j+a_3\mathbf k.

Given points $P1(x1,y1,z1)$, $P2(x2,y2,z2)$
P<em>1P</em>2→=⟨x<em>2−x</em>1,  y<em>2−y</em>1,  z<em>2−z</em>1⟩.\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 PQ→=⟨−1,5,3⟩,\overrightarrow{PQ}=\langle-1,5,3\rangle, length $\sqrt{35}$; unit vector ⟨−135,535,335⟩.\left\langle\tfrac{-1}{\sqrt{35}},\tfrac{5}{\sqrt{35}},\tfrac{3}{\sqrt{35}}\right\rangle.

Dot (Scalar) Product

For $\mathbf a=\langle a1,a2,a3\rangle$, $\mathbf b=\langle b1,b2,b3\rangle$:
a⋅b=a<em>1b</em>1+a<em>2b</em>2+a<em>3b</em>3.\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:
a⋅a=∥a∥2.\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⁡θ=a⋅b∥a∥ ∥b∥.\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: ∣a⋅b∣≤∥a∥ ∥b∥.|\mathbf a\cdot\mathbf b|\le\lVert\mathbf a\rVert\,\lVert\mathbf b\rVert.
Triangle: ∥a+b∥≤∥a∥+∥b∥.\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⁡α=a<em>1∥a∥,  cos⁡β=a</em>2∥a∥,  cos⁡γ=a3∥a∥,\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⁡2α+cos⁡2β+cos⁡2γ=1.\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1.

Projection and Component

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

Vector projection:
proj⁡<em>ab=a⋅b∥a∥2 a.\operatorname{proj}<em>{\mathbf a}\mathbf b = \frac{\mathbf a\cdot\mathbf b}{\lVert\mathbf a\rVert^{2}}\,\mathbf a. Any vector decomposes as b=proj⁡</em>ab+(b−proj⁡ab).\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=F⋅d.W=\mathbf F\cdot\mathbf d.

Cross (Vector) Product

For $\mathbf a=a1\mathbf i+a2\mathbf j+a3\mathbf k$ and $\mathbf b=b1\mathbf i+b2\mathbf j+b3\mathbf k$:
\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 ∥a×b∥=∥a∥ ∥b∥sin⁡θ.\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 τ=r×F,\boldsymbol\tau=\mathbf r\times\mathbf F, magnitude ∥τ∥=∥r∥ ∥F∥sin⁡θ.\lVert\boldsymbol\tau\rVert=\lVert\mathbf r\rVert\,\lVert\mathbf F\rVert\sin\theta.

Lines in Space

Given point $P0(x0,y0,z0)$ and direction vector $\mathbf v=\langle a,b,c\rangle$.

Parametric form:
x=x<em>0+at,  y=y</em>0+bt,  z=z0+ct.x=x<em>0+at,\;y=y</em>0+bt,\;z=z_0+ct.

Symmetric form (when $a,b,c\neq0$):
x−x<em>0a=y−y</em>0b=z−z0c.\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 $t1,t2$ 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=∥PQ→×v∥∥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 $P0(x0,y0,z0)$ with normal $\mathbf n=\langle a,b,c\rangle$ has equation
a(x−x<em>0)+b(y−y</em>0)+c(z−z0)=0.a(x-x<em>0)+b(y-y</em>0)+c(z-z_0)=0.

If two planes have normals $\mathbf n1,\mathbf n2$:
• They are parallel when $\mathbf n1\parallel\mathbf n2$.
• They are orthogonal when $\mathbf n1\cdot\mathbf n2=0$.

Angle $\theta$ between planes equals the angle between normals:
cos⁡θ=n<em>1⋅n</em>2∥n<em>1∥ ∥n</em>2∥.\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 $\mathbf n1\times\mathbf n2$.

Distance from point $P1$ to plane with normal $\mathbf n$ through $P0$:
D=∣n⋅P<em>1P</em>0→∣∥n∥.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.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=50.6x+10y+7z=50.

• Distance from $(1,-2,3)$ to line $x=3+t,y=-1-2t,z=4+t$ equals 356.\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 ∣a⋅(b×c)∣=27.|\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.