Linear_Algebra_Formula_Sheet

Linear Algebra & 3D Geometry — Complete Formula Sheet

From Zero to Planes & Distance

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1. Vectors — Basics

Vector Notation

$\vec{v}=⟨v_1,v_2,v_3⟩=v_1\vec{i}+v_2\vec{j}+v_3\vec{k}$

Magnitude (Length)

$|\vec{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$

Unit Vector

$\hat{v}=\frac{\vec{v}}{|\vec{v}|}$

Vector Between Two Points

$\overrightarrow{AB}=B-A=⟨b_1-a_1,b_2-a_2,b_3-a_3⟩$

⚠️ Direction = End point MINUS Start point. Do NOT use B’s coordinates as the direction vector!

Vector Addition & Scalar Multiplication

$\vec{u}+\vec{v}=⟨u_1+v_1,u_2+v_2,u_3+v_3⟩$
$c\vec{v}=⟨c{v}_1,c{v}_2,c{v}_3⟩$

Midpoint

$M=\frac{A+B}{2}$

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2. Dot Product

Definition

$\vec{u}\cdot \vec{v}=u_1{v}_1+u_2{v}_2+u_3{v}_3$
$\vec{u}\cdot \vec{v}=|\vec{u}||\vec{v}|\cos \theta$

Result is a SCALAR (number), not a vector.

Angle Between Two Vectors

$\cos \theta =\frac{\vec{u}\cdot \vec{v}}{|\vec{u}||\vec{v}|}$

Orthogonality Test

$\vec{u}\perp \vec{v}⟺\vec{u}\cdot \vec{v}=0$

Properties

• Commutative: $\vec{u}\cdot \vec{v}=\vec{v}\cdot \vec{u}$
• Distributive: $\vec{u}\cdot (\vec{v}+\vec{w})=\vec{u}\cdot \vec{v}+\vec{u}\cdot \vec{w}$
• $\vec{v}\cdot \vec{v}=|\vec{v}{|}^2$

Scalar Projection & Vector Projection

${\text{comp}}_{\vec{u}}(\vec{v})=\frac{\vec{u}\cdot \vec{v}}{|\vec{u}|}\text{[scalar — signed length of shadow]}$

${\text{proj}}_{\vec{u}}(\vec{v})=\frac{\vec{u}\cdot \vec{v}}{|\vec{u}{|}^2}\vec{u}\text{[vector — the shadow itself]}$

Work

$W=\vec{F}\cdot \vec{d}=|\vec{F}||\vec{d}|\cos \theta$

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3. Cross Product

Determinant Form

$\vec{u}\times \vec{v}=\left|\begin{array}{ccc}\vec{i}& \vec{j}& \vec{k}\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right|$

Component Form

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

⚠️ The j-component has a NEGATIVE sign in the determinant expansion! (middle term is subtracted)

Magnitude

$|\vec{u}\times \vec{v}|=|\vec{u}||\vec{v}|\sin \theta$

Key Properties

• Result is a VECTOR, perpendicular to both $\vec{u}$ and $\vec{v}$
• Anti-commutative: $\vec{u}\times \vec{v}=-(\vec{v}\times \vec{u})$
• NOT associative: $\vec{u}\times (\vec{v}\times \vec{w})\ne (\vec{u}\times \vec{v})\times \vec{w}$
• $\vec{u}\times \vec{u}=\vec{0}$
• Parallel test: $\vec{u}\times \vec{v}=\vec{0}⟺\vec{u}\parallel \vec{v}$

Geometric Applications

$\text{Area of parallelogram}=|\vec{u}\times \vec{v}|$
$\text{Area of triangle}=\frac{1}{2}|\vec{u}\times \vec{v}|$

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4. Scalar Triple Product & Parallelepiped

Scalar Triple Product

$\vec{u}\cdot (\vec{v}\times \vec{w})=\left|\begin{array}{ccc}{u}_1& u_2& u_3\\ v_1& v_2& v_3\\ w_1& w_2& w_3\end{array}\right|$

Volume of Parallelepiped (3D “Parallelogram”)

$V=|\vec{u}\cdot (\vec{v}\times \vec{w})|$

Three vectors $\vec{u},\vec{v},\vec{w}$ form the edges of a parallelepiped.
If $V=0$, the three vectors are coplanar (lie in the same plane).

Cyclic Property

$\vec{u}\cdot (\vec{v}\times \vec{w})=\vec{v}\cdot (\vec{w}\times \vec{u})=\vec{w}\cdot (\vec{u}\times \vec{v})$

Cyclic permutation preserves the value; swapping any two vectors flips the sign.

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5. Lines in 3D

Vector Equation

$\vec{r}(t)=\overrightarrow{r_0}+t\vec{d}=⟨x_0,y_0,z_0⟩+t⟨a,b,c⟩$

Parametric Equations

$x=x_0+at,y=y_0+bt,z=z_0+ct$

Symmetric Equations

$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$

If one component of $\vec{d}$ is 0 (say $a=0$), write: $x=x_0$, $\frac{y-y_0}{b}=\frac{z-z_0}{c}$

Line Through Two Points P, Q

$\vec{d}=Q-P,\text{then use }P\text{ as }\overrightarrow{r_0}$

Relationships Between Two Lines

• Parallel: $\overrightarrow{d_1}\times \overrightarrow{d_2}=\vec{0}$ (direction vectors are scalar multiples)
• Intersecting: Not parallel AND a common solution $(t,s)$ exists
• Skew: Not parallel AND no common solution — don’t lie in the same plane

Distance from Point Q to Line (through P with direction d)

$D=\frac{|\overrightarrow{PQ}\times \vec{d}|}{|\vec{d}|}$

Foot of Perpendicular from Q to Line

$F=P+t_0\vec{d},\text{where }{t}_0=\frac{\overrightarrow{PQ}\cdot \vec{d}}{\vec{d}\cdot \vec{d}}$

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6. Planes

Point-Normal Form

$\vec{n}\cdot (\vec{r}-\overrightarrow{r_0})=0$
$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$

$\vec{n}=⟨a,b,c⟩$ is the normal vector; $(x_0,y_0,z_0)$ is a point on the plane.

General Form

$ax+by+cz=d\text{where }d=a{x}_0+b{y}_0+c{z}_0$

Intercept Form

$\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1$

$\alpha ,\beta ,\gamma$ are the $x,y,z$ intercepts. Coefficient = 0 means plane is parallel to that axis.

Plane from Three Points P, Q, R

$\text{Step 1: }\overrightarrow{PQ}=Q-P,\overrightarrow{PR}=R-P$
$\text{Step 2: }\vec{n}=\overrightarrow{PQ}\times \overrightarrow{PR}$
$\text{Step 3: Use }\vec{n}\text{ and any of }P,Q,R\text{ in point-normal form}$

Finding Intercepts of $ax+by+cz=d$

• $x$-intercept: set $y=z=0\Rightarrow x=d/a$
• $y$-intercept: set $x=z=0\Rightarrow y=d/b$
• $z$-intercept: set $x=y=0\Rightarrow z=d/c$

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7. Line ↔ Plane Interactions

Intersection of Line and Plane

Substitute parametric equations into plane equation, solve for $t$.

• Unique $t$ → one intersection point
• $0=0$ (identity) → line lies in the plane
• $0=k$ (contradiction) → line is parallel, no intersection

Angle Between Line and Plane

$\sin \alpha =\frac{|\vec{d}\cdot \vec{n}|}{|\vec{d}||\vec{n}|}$

$\alpha$ = angle between line and plane (NOT the normal).
Equivalently: $\alpha =90°-\theta$, where $\theta$ = angle between $\vec{d}$ and $\vec{n}$.

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8. Two Planes

Angle Between Two Planes (Dihedral Angle)

$\cos \theta =\frac{|\overrightarrow{n_1}\cdot \overrightarrow{n_2}|}{|\overrightarrow{n_1}||\overrightarrow{n_2}|}$

Absolute value ensures $0°\le \theta \le 90°$.

Parallel Planes Test

$\overrightarrow{n_1}=k\overrightarrow{n_2}\text{ for some scalar }k$

If additionally $d_1=k\cdot d_2$, the planes are identical; otherwise parallel but distinct.

Line of Intersection of Two Planes

$\text{Direction vector: }\vec{d}=\overrightarrow{n_1}\times \overrightarrow{n_2}$
$\text{Find one point: set one variable}=0,\text{ solve the }2\times 2\text{ system}$
$\text{Parametric form: }\vec{r}=P+t\vec{d}$

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9. Distance Formulas ⭐

Point to Point

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

Point to Line

$D=\frac{|\overrightarrow{PQ}\times \vec{d}|}{|\vec{d}|}$

$P$ = point on line, $Q$ = external point, $\vec{d}$ = line direction.

Point to Plane

$D=\frac{|a{x}_1+b{y}_1+c{z}_1-d|}{\sqrt{a^2+b^2+c^2}}$

⚠️ SIGN WARNING: Plane is $ax+by+cz=d$. If $d=-5$, then: $|\text{value}-(-5)|=|\text{value}+5|$. Don’t drop the double negative!

Distance Between Parallel Planes

$ax+by+cz=d_1\text{and}ax+by+cz=d_2\text{(SAME }a,b,c\text{!)}$
$D=\frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}$

⚠️ Normalize BOTH planes to have identical LHS coefficients first!

Distance Between Skew Lines

$L_1:\vec{r}=\overrightarrow{a_1}+t\overrightarrow{d_1}{L}_2:\vec{r}=\overrightarrow{a_2}+s\overrightarrow{d_2}$

$D=\frac{|(\overrightarrow{a_2}-\overrightarrow{a_1})\cdot (\overrightarrow{d_1}\times \overrightarrow{d_2})|}{|\overrightarrow{d_1}\times \overrightarrow{d_2}|}$

Intuition: $\overrightarrow{d_1}\times \overrightarrow{d_2}$ is perpendicular to both lines; project the connector vector onto it.

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10. Quadric Surfaces Quick Reference

Surface	Standard Form	Key Feature
Sphere	$x^2+y^2+z^2=r^2$	All +, equal coefficients
Ellipsoid	$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$	All terms +, = 1
Hyperboloid (1 Sheet)	$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$	Two + one − = 1 (connected)
Hyperboloid (2 Sheets)	$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1$	Two + one − = −1 (disconnected)
Cone	$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$	Two + one − = 0 (vertex at origin)
Elliptic Paraboloid	$z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$	Linear = two + (bowl)
Hyperbolic Paraboloid	$z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$	Linear = + − (saddle)
Elliptic Cylinder	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	Missing variable → extends ∞
Parabolic Cylinder	$y=x^2$	Missing variable → extends ∞

Identification trick: Count positive/negative squared terms → check RHS (1, −1, or 0) → check if any variable is linear (not squared) → check if a variable is missing (cylinder).

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11. Common Pitfalls & Exam Tips ⚠️

Pitfall	Correction
Direction vector = $B$ (not $B-A$)	ALWAYS compute $B-A$
Cross product j-sign forgotten	j-component = $-(u_1{v}_3-u_3{v}_1)$. The minus is in front!
$5(x+1)=5x+1$	Distribute: $5(x+1)=5x+5$
Point-to-plane: $d$ is negative	If $d=-5$: $
Parallel planes: different coefficients	Normalize both to same $a,b,c$ first
Line-plane angle = $\arccos (\vec{d}\cdot \vec{n}/\dots )$	That’s angle with normal. Line-plane angle = $90°-$ that
Hyperboloid 1 vs 2 sheets	$=1$ → one sheet. $=-1$ → two sheets
Cone vs Hyperboloid	$=0$ → cone. $\ne 0$ → hyperboloid
”Eclipse”	It’s Ellipsoid, not Eclipse 😉

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Compiled for Sensei — CMU ‘30 — Multivariable Calculus Review