MCV4U · Exam review

Calculus and Vectors Exam Review

A practice exam with questions from all 6 units, then the key facts from every lesson, unit by unit. For a deeper look at one unit, open its test review.

Unit test reviews

Practice exam

36 questions, the same number from each unit. Each one shows the lesson it comes from.

Unit 1: Average and Instantaneous Rates of Change
1
Find the average rate of change of $f (x) = x^{3} - 2 x$ from $x = 1$ to $x = 3$ .
Unit 1: Average and Instantaneous Rates of Change
2
Find the average rate of change of $g (x) = 2^{x}$ from $x = 0$ to $x = 3$ . Give your answer to 2 decimal places.
Unit 1: Limits: Graphs, Tables, Sequences and Continuity
3
Match each sequence to its limit.
1. A. $π \approx 3.142$
2. B. $φ \approx 1.618$
3. C. $0$
4. D. $e \approx 2.718$
$(1 + \frac{1}{n})^{n}$
Ratio of consecutive Fibonacci numbers
Partial sums of $4 (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots)$
$\frac{1}{n}$
Unit 1: The Derivative from First Principles
4
Why can you not find $lim_{h \to 0} \frac{( 3 + h ) ^{2} - 9}{h}$ by substituting $h = 0$ right away?
Unit 1: The Derivative Function
5
If $f (x)$ is a cubic polynomial, what kind of function is $f^{'} (x)$ ?
Unit 1: The Derivative Function
6
For $f (x) = - 3 x + 7$ , what is $f^{'} (100)$ ?
Unit 2: Power, Constant, Sum and Difference Rules
7
What is $\frac{d}{d x} (x^{7})$ ?
Unit 2: The Product Rule
8
A ticket price is $p (t) = 25 + t$ dollars and the number of tickets sold per week is $n (t) = 800 - 20 t$ , with $t$ in weeks. Revenue is $R = p n$ . Find $R^{'} (5)$ in dollars per week.
dollars/week
Unit 2: The Chain Rule
9
Let $y = u^{3}$ and $u = x^{2} - 4 x$ . Find $\frac{d y}{d x}$ at $x = 1$ .
Unit 2: Derivatives of Rational Functions
10
For $f (x) = \frac{x + 1}{x - 5}$ , where does $f^{'} (x)$ fail to exist?
Unit 2: Derivatives of Sine and Cosine
11
Why must $x$ be in radians for $\frac{d}{d x} sin x = cos x$ to hold?
Unit 2: Derivatives of Exponential Functions
12
For $f (x) = a^{x}$ ( $a > 0$ , $a \neq = 1$ ), what can you say about the ratio $\frac{f ^{'} ( x )}{f ( x )}$ ?
Unit 3: Increasing and Decreasing Functions and the First Derivative Test
13
On the interval $(2, 5)$ , $f^{'} (x) > 0$ . What does this tell you about $f$ ?
Unit 3: Increasing and Decreasing Functions and the First Derivative Test
14
What is the name for a number $c$ in the domain of $f$ where $f^{'} (c) = 0$ or $f^{'} (c)$ does not exist? (Two words.)
Unit 3: Concavity, Points of Inflection and the Second Derivative Test
15
A function has $f^{'} (4) = 0$ and $f^{''} (4) = - 5$ . What can you conclude?
Unit 3: Sketching Polynomial and Simple Rational Functions
16
Find the maximum value of $f (x) = \frac{4 x}{x ^{2} + 1}$ .
Unit 3: Graphs of f, f′ and f″
17
At a local maximum of a smooth function $f$ , what does the graph of $f^{'}$ do?
Unit 3: Graphs of f, f′ and f″
18
Match each feature of the graph of $f^{'}$ with the matching feature of $f$ .
1. A. $f$ is concave up
2. B. $f$ is decreasing
3. C.Local minimum of $f$
4. D.Point of inflection of $f$ (concave up to concave down)
$f^{'}$ crosses the axis from $-$ to $+$
$f^{'}$ has a local maximum
$f^{'}$ is below the axis
$f^{'}$ is increasing
Unit 4: Motion: Displacement, Velocity and Acceleration
19
At some instant an object has $v = - 3$ m/s and $a = - 2 m/s^{2}$ . Which statement is true?
Unit 4: Motion: Displacement, Velocity and Acceleration
20
The derivative of velocity with respect to time is called what?
Unit 4: Rates of Change in Context
21
A tank holds $V (t) = 1000 (1 - \frac{t}{40})^{2}$ litres of water as it drains, $0 \leq t \leq 40$ minutes. Find $V^{'} (10)$ in litres per minute.
L/min
Unit 4: Optimization Problems
22
Two positive numbers $x$ and $y$ add to $30$ . Find the maximum value of $x y^{2}$ .
Unit 4: Optimization in Business and Science
23
At a price of $p$ dollars, $q = 1200 - 40 p$ tickets sell. What price, in dollars, maximizes revenue?
dollars
Unit 4: Optimization in Business and Science
24
What is the derivative of a revenue function, $R^{'} (x)$ , called? (Two words.)
Unit 5: Vectors, Scalars and Direction
25
A hiker is known to be $69$ km from Lindsay, Ontario. Why is that not enough to locate the hiker?
Unit 5: Adding, Subtracting and Scaling Vectors
26
Simplify $2 (a + 3 b) - 3 (a - b)$ .
Unit 5: Cartesian Vectors in 2-Space and 3-Space
27
Find the distance between $P (- 1, 4, 2)$ and $Q (1, 0, 6)$ .
Unit 5: Applications: Forces, Wind and Current
28
Match each navigation term to its meaning.
Heading
Track
Air speed
Ground speed
Unit 5: The Dot Product
29
A rope pulls a sled $15$ m across flat snow with a force of $120$ N at $25°$ above the horizontal. How much work does the rope do, to the nearest joule?
J
Unit 5: The Cross Product
30
What rule do you use to find the direction of $a \times b$ ?
Unit 6: Lines in 2-Space
31
Which vector is a normal to the line $5 x - 2 y + 7 = 0$ ?
Unit 6: Lines in 3-Space
32
Why can't a line in 3-space be described by one scalar equation?
Unit 6: Planes: Normals and Equations
33
Which vector is a normal to the plane $3 x + 5 y - 2 z = 6$ ?
Unit 6: Planes: Normals and Equations
34
Put the steps for finding the scalar equation of a plane through three points $A$ , $B$ and $C$ in order.
1. 1Check that B and C also satisfy the equation
2. 2Substitute point A to solve for D
3. 3Write Ax + By + Cz + D = 0 using that normal
4. 4Form the vectors AB and AC
5. 5Take their cross product to get a normal (A, B, C)
Unit 6: Intersections of Lines and Planes
35
Sort these arrangements of three planes by the size of their common solution set, from smallest to largest.
1. 1Three planes whose normals are not coplanar
2. 2Three coincident planes
3. 3Three planes through one common line, like pages of a book
4. 4Three parallel, distinct planes
Unit 6: Distances in 3-Space
36
Match each distance to the formula used to find it.
1. A. $\frac{∣ P _{0} Q \times m ∣}{∣ m ∣}$
2. B. $\frac{∣ P _{1} P _{2} \cdot ( m _{1} \times m _{2} ) ∣}{∣ m _{1} \times m _{2} ∣}$
3. C. $\frac{∣ A x _{1} + B y _{1} + C ∣}{A ^{2} + B ^{2}}$
4. D. $\frac{∣ A x _{1} + B y _{1} + C z _{1} + D ∣}{A ^{2} + B ^{2} + C ^{2}}$
Point to plane
Point to line in 2-space
Point to line in 3-space
Between skew lines

Key facts by unit

Unit 1: Rates of Change & Limits

Average and Instantaneous Rates of Change

Rate of change $= \frac{Δ y}{Δ x}$ , with units of output units per input unit (m/s, L/min, °C/km).
Average rate of change from $a$ to $b$ : $\frac{f ( b ) - f ( a )}{b - a}$ , the slope of the secant through $(a, f (a))$ and $(b, f (b))$ .
Instantaneous rate of change at $x = a$ : the slope of the tangent at $(a, f (a))$ .
Estimate a tangent slope with secants through the point: shrink the interval from both sides and see what value the slopes approach.
A graph with a corner at a point has no tangent there, so its instantaneous rate of change at that point does not exist.

Limits: Graphs, Tables, Sequences and Continuity

$lim_{x \to a} f (x) = L$ means $f (x)$ gets as close to $L$ as you like when $x$ is close enough to $a$ (but not equal to it).
The two-sided limit exists only if $lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) .$
If substitution gives $\frac{0}{0}$ , simplify (factor and cancel) and then substitute.
Famous limits: $(1 + \frac{1}{n})^{n} \to e \approx 2.718$ , Fibonacci ratios $\to φ \approx 1.618$ , $4 (1 - \frac{1}{3} + \frac{1}{5} - \dots) \to π$ .
$f$ is continuous at $a$ if $f (a)$ is defined, the limit exists, and $lim_{x \to a} f (x) = f (a)$ .

The Derivative from First Principles

Difference quotient: $\frac{f ( a + h ) - f ( a )}{h}$ , the slope of the secant from $x = a$ to $x = a + h$ (an average rate of change).
Derivative at a point: $f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h},$ the slope of the tangent (an instantaneous rate of change).
You cannot substitute $h = 0$ directly ( $\frac{0}{0}$ ); expand, simplify and cancel the factor of $h$ first.
For $f (x) = x^{2}$ at $x = 3$ : the quotient simplifies to $6 + h$ , so $f^{'} (3) = 6$ .
$f^{'} (a)$ has units of (output units) per (input unit), such as m/s.

The Derivative Function

Derivative function: $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h};$ it gives the tangent slope at every $x$ .
Notation: $f^{'} (x)$ , $y^{'}$ , $\frac{d y}{d x}$ and $\frac{d}{d x} f (x)$ all mean the derivative.
For polynomials the degree drops by one: linear gives constant, quadratic gives linear, cubic gives quadratic.
Where $f$ rises, $f^{'} (x) > 0$ ; where $f$ falls, $f^{'} (x) < 0$ (apart from single flat points, like $x^{3}$ at $0$ ); a smooth local maximum or minimum has $f^{'} (x) = 0$ .
No derivative at a corner, cusp, discontinuity or vertical tangent. Differentiable implies continuous, but not the reverse.

Unit 2: Derivative Rules

Power, Constant, Sum and Difference Rules

Power rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ for natural and rational exponents, including negative ones.
Constant rule: $\frac{d}{d x} (c) = 0$ . Constant multiple rule: $\frac{d}{d x} [k f (x)] = k f^{'} (x)$ .
Sum and difference rules: $(f \pm g)^{'} (x) = f^{'} (x) \pm g^{'} (x)$ , so polynomials are differentiated term by term.
Rewrite before differentiating: $x = x^{\frac{1}{2}}$ and $\frac{1}{x ^{n}} = x^{- n}$ .
Tangent slope at $x = a$ is $f^{'} (a)$ ; the tangent line is $y - f (a) = f^{'} (a) (x - a)$ .
To find where the slope equals $m$ , solve $f^{'} (x) = m$ , then use $f (x)$ to find each point's $y$ -coordinate.

The Product Rule

Product rule: $(f g)^{'} = f^{'} g + f g^{'}$ .
$(f g)^{'} \neq = f^{'} g^{'}$ : check with $f = g = x$ , where $(x^{2})^{'} = 2 x$ but $1 \cdot 1 = 1$ .
Leibniz form: $\frac{d}{d x} (uv) = \frac{d u}{d x} v + u \frac{d v}{d x}$ .
For three factors: $(f g h)^{'} = f^{'} g h + f g^{'} h + f g h^{'}$ .
To find $f^{'} (a)$ , substitute $a$ into each of $f, g, f^{'}, g^{'}$ ; there is no need to simplify first.

The Chain Rule

Chain rule (Leibniz): $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$ .
Chain rule (function notation): $[f (g (x))]^{'} = f^{'} (g (x)) \cdot g^{'} (x) .$
Power of a function: $\frac{d}{d x} [g (x)]^{n} = n [g (x)]^{n - 1} g^{'} (x) .$
Radical: $\frac{d}{d x} g (x) = \frac{g ^{'} ( x )}{2 g ( x )}$ .
Linear inside: $\frac{d}{d x} (a x + b)^{n} = an (a x + b)^{n - 1}$ ; the $a$ is easy to forget.

Derivatives of Rational Functions

Quotient rule: $(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$ , where $g \neq = 0$ .
Alternative: $\frac{f}{g} = f \cdot g^{- 1}$ , then use the product and chain rules.
Constant numerator: $\frac{d}{d x} [\frac{k}{g ( x )}] = - \frac{k g ^{'} ( x )}{[ g ( x ) ] ^{2}} .$
$(\frac{f}{g})^{'} \neq = \frac{f ^{'}}{g ^{'}}$ , and the numerator order $f^{'} g - f g^{'}$ matters.
A rational function's derivative does not exist where its denominator is $0$ ; horizontal tangents occur where the numerator of $f^{'}$ is $0$ .

Derivatives of Sine and Cosine

$\frac{d}{d x} (sin x) = cos x$ and $\frac{d}{d x} (cos x) = - sin x$ , with $x$ in radians.
Chain rule versions: $\frac{d}{d x} sin (g (x)) = cos (g (x)) g^{'} (x)$ and $\frac{d}{d x} cos (g (x)) = - sin (g (x)) g^{'} (x) .$
$\frac{d}{d x} sin (k x) = k cos (k x)$ and $\frac{d}{d x} cos (k x) = - k sin (k x)$ .
$\frac{d}{d x} tan x = \frac{1}{cos ^{2} x} = sec^{2} x$ (quotient rule plus $sin^{2} x + cos^{2} x = 1$ ).
Repeated derivatives of $sin x$ cycle: $cos x, - sin x, - cos x, sin x$ .

Derivatives of Exponential Functions

$\frac{d}{d x} (e^{x}) = e^{x}$ : for $y = e^{x}$ , slope equals height at every point.
$\frac{d}{d x} (a^{x}) = a^{x} ln a$ for $a > 0$ , $a \neq = 1$ ; so $\frac{f ^{'} ( x )}{f ( x )} = ln a$ is constant.
Chain rule with $e$ : $\frac{d}{d x} e^{g (x)} = e^{g (x)} g^{'} (x)$ , for example $\frac{d}{d x} e^{k x} = k e^{k x}$ .
$ln x = lo g_{e} x$ is the inverse of $e^{x}$ : $e^{l n x} = x$ (for $x > 0$ ) and $ln (e^{x}) = x$ .
The graphs of $y = e^{x}$ and $y = ln x$ are reflections in $y = x$ ; $(0, 1) \leftrightarrow (1, 0)$ and $(1, e) \leftrightarrow (e, 1)$ .
$e \approx 2.71828$ is the base between $2$ and $3$ whose tangent slope at $x = 0$ is exactly $1$ .

Unit 3: Curve Sketching

Increasing and Decreasing Functions and the First Derivative Test

If $f^{'} (x) > 0$ on an interval, $f$ is increasing there; if $f^{'} (x) < 0$ , $f$ is decreasing there.
A critical number is a $c$ in the domain of $f$ with $f^{'} (c) = 0$ or $f^{'} (c)$ undefined.
First derivative test: $f^{'}$ changes $+$ to $-$ at $c$ gives a local maximum; $-$ to $+$ gives a local minimum; no sign change gives neither.
$f^{'} (c) = 0$ alone does not guarantee an extremum: $f (x) = x^{3}$ has $f^{'} (0) = 0$ but no maximum or minimum at $0$ .
Find the height of an extremum from $f (c)$ , not $f^{'} (c)$ .

Concavity, Points of Inflection and the Second Derivative Test

$f^{''} (x)$ is the derivative of $f^{'} (x)$ : the rate of change of the slope of the tangent.
$f^{''} (x) > 0$ means concave up (slopes increasing); $f^{''} (x) < 0$ means concave down (slopes decreasing).
A point of inflection is a point on the graph where the concavity changes; check that $f^{''}$ changes sign, since $f^{''} (c) = 0$ alone is not enough (e.g. $x^{4}$ at $0$ ).
At an inflection point where $f^{'}$ exists, $f^{'}$ has a local maximum or minimum.
Second derivative test: $f^{'} (c) = 0$ and $f^{''} (c) > 0$ gives a local minimum; $f^{''} (c) < 0$ gives a local maximum; $f^{''} (c) = 0$ is inconclusive.

Sketching Polynomial and Simple Rational Functions

Sketching order: domain, intercepts, symmetry, $f^{'}$ (critical numbers, increase and decrease, extrema), $f^{''}$ (concavity, inflection points), asymptotes and end behaviour, then sketch and verify.
Even: $f (- x) = f (x)$ , symmetric about the $y$ -axis. Odd: $f (- x) = - f (x)$ , symmetric about the origin.
A degree- $n$ polynomial has at most $n - 1$ turning points and at most $n - 2$ points of inflection.
For a rational function, vertical asymptotes go on the sign charts as boundaries, but they are never critical numbers or inflection points.
Horizontal asymptote: $y = 0$ if the degree of the numerator is smaller; the ratio of leading coefficients if the degrees are equal.

Graphs of f, f′ and f″

$f^{'} > 0$ on an interval $\to$ $f$ increasing (a rising $f$ can still have $f^{'} = 0$ at a single flat point); local max or min of $f$ $\leftrightarrow$ $f^{'}$ crosses zero ( $+$ to $-$ or $-$ to $+$ ).
$f^{''} > 0$ on an interval $\to$ $f^{'}$ increasing $\leftrightarrow$ $f$ concave up (a concave-up $f$ can still have $f^{''} = 0$ at a single point, like $x^{4}$ at $0$ ).
Inflection point of $f$ $\leftrightarrow$ local max or min of $f^{'}$ $\leftrightarrow$ $f^{''}$ changes sign.
$f (x) + k$ has the same derivative as $f (x)$ ; the derivative of $f (x - h)$ is $f^{'} (x - h)$ .
Knowing $f^{'}$ determines $f$ only up to a vertical shift: $f (x) + C$ for any constant $C$ . One known point fixes $C$ .

Unit 4: Applications of Derivatives

Motion: Displacement, Velocity and Acceleration

$v (t) = s^{'} (t)$ and $a (t) = v^{'} (t) = s^{''} (t)$ ; speed is $∣ v (t) ∣$ .
At rest: $v (t) = 0$ . Changes direction: $v$ changes sign.
Speeding up when $v$ and $a$ have the same sign; slowing down when they have opposite signs.
Displacement $= s (end) - s (start)$ . Total distance adds $∣Δ s ∣$ over each piece between turning points.
For $h (t) = - 4.9 t^{2} + v_{0} t + h_{0}$ : maximum height where $h^{'} (t) = 0$ , landing where $h (t) = 0$ .

Rates of Change in Context

The units of $Q^{'} (t)$ are (units of $Q$ ) per (unit of $t$ ); the sign says increasing or decreasing.
$\frac{d}{d t} b^{t} = b^{t} ln b$ and $\frac{d}{d t} e^{k t} = k e^{k t}$ : exponential growth and decay rates are proportional to the amount present.
For a sinusoidal model $A sin (k (t - c)) + d$ , the rate is $A k cos (k (t - c))$ (radians); it is fastest at the midline and zero at a maximum or minimum.
Inflation: $P^{'} (t) > 0$ ; deflation: $P^{'} (t) < 0$ ; inflation is fastest where $P^{''} (t) = 0$ and $P^{''}$ changes from positive to negative.
Relative (percent) rate of change: $\frac{Q ^{'} ( t )}{Q ( t )} \times 100%$ .

Optimization Problems

On a closed interval, the absolute maximum and minimum occur at critical numbers or endpoints: evaluate them all and compare.
Use the constraint to write the quantity to optimize as a function of ONE variable, then state its domain.
Reject critical numbers outside the domain; confirm a max or min with the first or second derivative test.
Open box from an $L \times W$ sheet: $V (x) = x (L - 2 x) (W - 2 x)$ with $0 < x < \frac{min ( L , W )}{2}$ .
To minimize a distance, minimize $d^{2}$ instead; the same $x$ gives the minimum.

Optimization in Business and Science

Revenue $R = (price) (quantity sold)$ ; profit $P (x) = R (x) - C (x)$ .
Marginal revenue is $R^{'} (x)$ and marginal cost is $C^{'} (x)$ ; a maximum profit inside the domain occurs where $R^{'} (x) = C^{'} (x)$ .
Average cost is $\frac{C ( x )}{x}$ ; differentiate it (it is a rational function) to find the lowest cost per item.
For $R (x) = k x b^{- x}$ with $k > 0$ and $b > 1$ , the maximum is at $x = \frac{1}{ln b}$ ; for the bus fare, $\frac{1}{ln 1.15} \approx 7.16$ .
Compare numeric (table), graphical and algebraic (derivative) strategies: the algebraic one gives the exact value.
Always answer in context with units, a domain check and sensible rounding.

Unit 5: Vectors

Vectors, Scalars and Direction

A scalar has magnitude only; a vector has magnitude and direction. Magnitude $∣ v ∣ \geq 0$ .
Equal vectors have the same magnitude and direction, regardless of position. $B A = - A B$ .
Standard angle: counterclockwise from the positive $x$ -axis. True bearing: clockwise from north, three digits.
Convert between them with $θ = 90° - β$ (add $360°$ if negative).
Quadrant bearing: start at N or S, turn toward E or W, angle from $0°$ to $90°$ (for example, a true bearing of $320°$ is N $40°$ W).

Adding, Subtracting and Scaling Vectors

Triangle law: place the tail of $b$ on the head of $a$ ; $a + b$ runs from the first tail to the last head.
$a - b = a + (- b)$ ; tail to tail, it runs from the head of $b$ to the head of $a$ .
$∣ k a ∣ = ∣ k ∣∣ a ∣$ ; $k < 0$ reverses the direction. Non-zero $a$ and $b$ are collinear when $b = k a$ .
With $θ$ the angle between them (tail to tail): $∣ a \pm b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} \pm 2∣ a ∣∣ b ∣ cos θ .$
Addition is commutative and associative; scalar multiplication distributes over both vector and scalar addition.

Cartesian Vectors in 2-Space and 3-Space

$A B =$ head minus tail $= (x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}) .$
$∣ (x, y, z) ∣ = x^{2} + y^{2} + z^{2}$ ; in 2-space drop $z$ .
From magnitude $r$ and standard angle $θ$ : $x = r cos θ$ , $y = r sin θ$ .
From components: $r = x^{2} + y^{2}$ , $θ = tan^{- 1} (y / x)$ , plus $180°$ if $x < 0$ , plus $360°$ if $x > 0$ and $y < 0$ .
Add, subtract and scale component by component; $i = (1, 0, 0)$ , $j = (0, 1, 0)$ , $k = (0, 0, 1)$ .

Applications: Forces, Wind and Current

Resultant $R = F_{1} + F_{2} + \dots$ ; equilibrant $E = - R$ .
Ground velocity $=$ air (or water) velocity $+$ wind (or current) velocity.
A wind FROM the south blows TOWARD the north; a current is named by the way it flows.
For a heading measured from north, east component $= v sin β$ , north component $= v cos β$ .
Crossing time $= \frac{width}{across component}$ ; to land directly opposite, aim upstream at $α$ with $sin α = \frac{v _{current}}{v _{boat}}$ .

The Dot Product

$a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3},$ a scalar.
$cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$ ; non-zero vectors are orthogonal exactly when $a \cdot b = 0$ .
$a \cdot a = ∣ a ∣^{2}$ ; the dot product is commutative and distributive, and scalars factor out.
Scalar projection of $a$ onto $b$ : $\frac{a \cdot b}{∣ b ∣}$ . Vector projection: $\frac{a \cdot b}{∣ b ∣ ^{2}} b$ .
Work $W = F \cdot d = ∣ F ∣∣ d ∣ cos θ$ , in joules.

The Cross Product

$a \times b = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1}),$ perpendicular to both $a$ and $b$ (right-hand rule).
$∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$ = area of the parallelogram on $a$ and $b$ ; a triangle is half.
$b \times a = - (a \times b)$ ; collinear vectors give $0$ ; distributive, but not associative.
Torque $τ = r \times F$ , $∣ τ ∣ = ∣ r ∣∣ F ∣ sin θ$ (N·m), largest at $θ = 90°$ .
Volume of a parallelepiped $= ∣ a \cdot (b \times c) ∣$ ; zero means the three vectors are coplanar.

Unit 6: Lines & Planes

Lines in 2-Space

Scalar equation of a line in 2-space: $A x + B y + C = 0$ , with normal $n = (A, B)$ .
Vector equation: $r = r_{0} + t m$ , $t \in R$ ; parametric: $x = x_{0} + t m_{1}$ , $y = y_{0} + t m_{2}$ .
From normal $(A, B)$ , a direction is $(B, - A)$ ; from direction $(m_{1}, m_{2})$ , a normal is $(m_{2}, - m_{1})$ .
Slope $= \frac{m _{2}}{m _{1}}$ (when $m_{1} \neq = 0$ ) $= - \frac{A}{B}$ (when $B \neq = 0$ ).
Two lines in 2-space: one point if the normals are not parallel; otherwise coincident (infinitely many) or parallel and distinct (none).

Lines in 3-Space

A single equation $A x + B y + C z + D = 0$ is a plane; a line in 3-space needs two equations (two intersecting planes).
Vector: $r = r_{0} + t m$ ; parametric: $x = x_{0} + t m_{1}$ , $y = y_{0} + t m_{2}$ , $z = z_{0} + t m_{3}$ .
Symmetric: $\frac{x - x _{0}}{m _{1}} = \frac{y - y _{0}}{m _{2}} = \frac{z - z _{0}}{m _{3}}$ (each $m_{i} \neq = 0$ ; a zero component makes that coordinate constant).
Parallel directions: coincident or parallel and distinct. Non-parallel directions: intersecting or skew; the third equation decides.
Angle between lines: $cos θ = \frac{∣ m _{1} \cdot m _{2} ∣}{∣ m _{1} ∣∣ m _{2} ∣};$ perpendicular when $m_{1} \cdot m_{2} = 0$ .

Planes: Normals and Equations

Scalar equation of a plane: $A x + B y + C z + D = 0$ , with normal $n = (A, B, C)$ .
Through $P_{0}$ with normal $n$ : $A (x - x_{0}) + B (y - y_{0}) + C (z - z_{0}) = 0,$ so $D = - (A x_{0} + B y_{0} + C z_{0})$ .
Vector equation: $r = r_{0} + s a + t b$ , with $a$ and $b$ non-parallel vectors in the plane.
Vector to scalar: $n = a \times b$ ; three points $A$ , $B$ , $C$ : $n = A B \times A C$ .
Parallel planes: parallel normals. Perpendicular planes: $n_{1} \cdot n_{2} = 0$ .

Intersections of Lines and Planes

Line meets plane: substitute the parametric equations into the scalar equation and solve for $t$ . If $n \cdot m = 0$ the line is parallel to the plane or lies in it.
Two non-parallel planes meet in a line with direction $n_{1} \times n_{2}$ .
Three planes meet at exactly one point if and only if $n_{1} \cdot (n_{2} \times n_{3}) \neq = 0$ .
Elimination giving $0 = 0$ : infinitely many solutions (a line or a plane). Giving $0 = k$ , $k \neq = 0$ : no solution.
Two equations in 2-space usually give a point; two equations in 3-space usually give a line.

Distances in 3-Space

Point to plane: $d = \frac{∣ A x _{1} + B y _{1} + C z _{1} + D ∣}{A ^{2} + B ^{2} + C ^{2}}$ (in 2-space, $\frac{∣ A x _{1} + B y _{1} + C ∣}{A ^{2} + B ^{2}}$ ).
Parallel planes with the same $A$ , $B$ , $C$ : $d = \frac{∣ D _{1} - D _{2} ∣}{A ^{2} + B ^{2} + C ^{2}}$ .
Point to line in 3-space: $d = \frac{∣ P _{0} Q \times m ∣}{∣ m ∣}$ .
Skew lines: $d = \frac{∣ P _{1} P _{2} \cdot ( m _{1} \times m _{2} ) ∣}{∣ m _{1} \times m _{2} ∣};$ $d = 0$ means the lines intersect.
The foot of the perpendicular from $Q$ to a plane is where the line $r = OQ + k n$ meets the plane.