Intuition The ONE core ideaA line in 3D is nothing more than one spot you stand on plus one arrow saying which way to walk — take any number of steps forward or backward along that arrow and you sweep out the whole line. Every symbol on the parent page (r , a , b , λ , x 1 , a , b , c
Before you can read the parent note the line-equation page , you must be fluent in a handful of tiny pieces. We build each one from absolute zero, in the order that lets the next piece stand on the one before.
Everything happens inside a room . To name any spot in the room we need three numbers , because the room has three independent directions: left↔right, forward↔back, up↔down.
Definition Coordinates and axesWe draw three number-lines through one corner called the origin O
x y z
A point is a spot in the room, written ( x , y , z ) x x y y z ( 0 , 0 , 0 )
Why the topic needs it: a line is a set of points , so we cannot talk about lines until we can name a single point with three numbers.
A vector is the star of the whole topic. Picture an arrow: it has a length (how far) and a direction (which way). Crucially, a vector does not care where it starts — only its length and heading matter.
Definition Vector and its componentsA vector in 3D is written b = ( a , b , c ) b = a i ^ + b j ^ + c k ^ components : how far the arrow reaches along x y z
i ^ , j ^ , k ^ unit arrows : length-1 arrows pointing purely along x y z So 2 i ^ + 3 j ^ − k ^ x y backward along z
Intuition Two flavours of arrow — and why we must keep them straightThe same arrow ( a , b , c ) plays two different jobs in this topic:
Position vector — its tail is nailed to the origin O is a point. This is how we turn a point into an arrow.Direction vector — a free-floating arrow that only says "this way." We are allowed to slide it anywhere and even stretch it.
The parent note calls the anchor's position vector a b
Why the topic needs it: the parent's master equation r = a + λ b all vectors — a position arrow, a direction arrow, and a general position arrow r
Two operations on arrows power everything.
Definition Scalar multiplicationA scalar is just an ordinary number (a "scaler" — it scales things). Multiplying a vector by a scalar λ λ λ ( a , b , c ) = ( λa , λb , λ c ) . λ = 2 λ = − 1 λ = 0
Definition Vector addition (tip-to-tail)To add a + d d tail on a tip ; the sum is the arrow from the very first tail to the very last tip. In components you just add matching numbers:
( a 1 , a 2 , a 3 ) + ( d 1 , d 2 , d 3 ) = ( a 1 + d 1 , a 2 + d 2 , a 3 + d 3 ) .
Why the topic needs it: r = a + λ b a λ b λ Vectors — scalar multiplication & parallel vectors .
Here is the hinge of the entire subtopic.
Definition Parallel vectorsTwo vectors are parallel when they point along the same straight track — one is just a scalar multiple of the other:
u ∥ v ⟺ u = λ v for some number λ . ( 2 , 3 , − 1 ) ( 4 , 6 , − 2 ) 2 ( 2 , 3 , − 1 ) ( − 2 , − 3 , 1 ) − 1 same line ).
Intuition Why "parallel" builds the lineStand at anchor A P A P must lie along the line's track — so it must be parallel to the direction b A P = λ b is the line equation; the parent note just rearranges it.
Why the topic needs it: it explains why the parent writes A P = λ b
We often need the arrow from one point to another .
Common mistake Which way round?Wrong instinct: A P = a − r A Fix: the arrow points towards its second letter P P minus A r − a
Why the topic needs it: combining §3 and §4 gives r − a = λ b ⇒ r = a + λ b
λ lambda ) is a free number we are allowed to choose. Think of it as a dial measuring how many "b
λ = 0 A λ = 1 A + b λ = − 2 b backward .Every real number λ ∈ R λ
Intuition Why ONE dial controls all three coordinatesWhen you turn the single dial λ x , y , z r = a + λ b x = x 1 + λa , y = y 1 + λb , z = z 1 + λ c same λ one value of λ all three equations (Example 4 on the parent page).
Why the topic needs it: λ λ
Definition Direction ratiosThe components a , b , c b direction ratios (DRs) of the line. They answer: "for each step along the line, how much x y z unique only up to a nonzero scale : ( 2 , 3 , − 1 ) ( 4 , 6 , − 2 ) ( − 2 , − 3 , 1 )
R λ ∈ R λ you may walk any distance, either way.
Why the topic needs it: DRs are the denominators in symmetric form and the whole content of Direction ratios and direction cosines .
Symbol
Plain words
Picture
O origin
corner where axes meet
( x , y , z ) a point
a spot in the room
i ^ , j ^ , k ^ unit arrows
length-1 arrows along axes
a position vector of anchor A
arrow origin→A
b = ( a , b , c ) direction vector / DRs
free arrow: the line's heading
r = ( x , y , z ) general point's position
arrow origin→"any point P
λ parameter
steps-along-b
A P = r − a displacement A P
arrow between two points
Components and unit arrows
Scalar multiply and stretch
Parallel means scalar multiple
Line equation r = a + lambda b
Parameter lambda the steps dial
Direction ratios feed symmetric form
Name three numbers needed to locate any point in 3D and why The coordinates ( x , y , z )
What does the vector 2 i ^ + 3 j ^ − k ^ An arrow reaching 2 along x y z
What does multiplying a vector by λ = − 1 Same length, flipped to point the opposite way.
State the tip-to-tail rule for adding two vectors Put the second arrow's tail on the first's tip; the sum runs from the first tail to the last tip (components add).
When are two vectors parallel? When one is a scalar multiple of the other,
u = λ v .
Write the displacement arrow from A a P r A P = r − a (destination minus start).
What does λ = 0 λ = − 2 λ = 0 A λ = − 2
Are direction ratios unique? No — only up to a nonzero scalar multiple.
Why must one value of λ Because the single dial
λ controls
x , y , z together in
r = a + λ b .
What does λ ∈ R λ
