A line in space is a point plus any multiple of a direction, r = a + λb; a plane is a point plus a normal, r·n = c, and that normal comes from the vector product of two directions lying in the plane. Two lines that are not parallel still need not meet: setting r₁ = r₂ gives three component equations in two unknowns, and only if the third equation agrees with the pair found from the first two do the lines intersect. A line meets a plane where substituting a + λb into r·n = c leaves one value of λ. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.
A line is one point plus every scalar multiple of a direction; a plane is one point plus every combination of two. That's the whole topic: fix where you start, then say which way you're allowed to move. Every question is just asking where two of these freedoms cross, miss, or coincide.
Line 1, r₁ = (2,1,3) + λ₁(1,1,2), drawn with a second line that meets it, runs parallel to it or is skew to it, or with a plane and its normal; changing λ₁ and λ₂ shows the gap between the two moving points closing only where the lines truly meet, and the single λ₁ where line 1 pierces the plane −y + z = 4.
knowhere maps every concept to what it rests on and what rests on it — 4 underneath this one, 0 built on top. Each one says why, in a sentence, not as an arrow on a diagram.
Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.
Every HSC and VCE concept in knowhere is built like this one — mapped to your curriculum, never capped by it. Free for a week. No lock-in, cancel inside the week and pay nothing.
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