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Formula for the angle between two lines?
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All Flashcards in Topic 3.15
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3.15.18 cards
Formula for the angle between two lines?
cos θ = |d₁·d₂| / (|d₁| |d₂|), where d₁, d₂ are the direction vectors.
Why does the angle between two lines use only the direction vectors?
Sliding a line (keeping its direction) doesn't change the angle, so the base points are irrelevant — only the directions matter.
Why the absolute-value bars in cos θ = |d₁·d₂|/(|d₁||d₂|)?
They keep cos θ positive so you report the ACUTE angle; a negative dot product would otherwise give an obtuse angle.
How do you test whether two lines are perpendicular?
Show their direction vectors have dot product zero: d₁·d₂ = 0 ⟺ perpendicular.
Angle between directions (1, −1, 2) and (2, 1, 1)?
Dot = 3, |d₁| = |d₂| = √6, cos θ = 3/6 = ½ ⇒ θ = 60°.
If d₁·d₂ is negative, what does that tell you?
The arrows make an obtuse angle; take the modulus to get the acute angle between the lines.
Do the lengths of the direction vectors change the angle?
No — the formula divides by both magnitudes, so any scaling of a direction cancels out.
What is the denominator in the angle formula?
The PRODUCT of the magnitudes |d₁| × |d₂| (not their sum).
3.15.28 cards
What are the three ways two lines can sit in 3D?
Parallel (same direction), intersecting (meet at one point), or skew (not parallel and never meet).
How do you check if two lines are parallel?
See if one direction vector is a scalar multiple of the other: d₂ = k·d₁.
What is a skew pair of lines?
Lines that are NOT parallel and yet NEVER meet — only possible in 3D.
How do you find the intersection of two lines?
Equate the position vectors (3 component equations), solve two for s and t, then test the third; if it holds, sub s back for the point.
Why solve only two of the three equations?
Two equations fix s and t; the third is the consistency check — it tells you whether the lines actually meet.
How do you prove two lines are skew?
Show the directions are NOT parallel AND the equation system is inconsistent (no common s, t).
Is 'the lines never meet' enough to call them skew?
No — parallel lines also never meet. You must also show the directions are not parallel.
Lines perpendicular and intersecting: how do you find unknown constants?
Perpendicularity (dot product = 0) gives a direction unknown; forcing the intersection (third equation) gives a position unknown.
Topic 3.15 study notes
Full notes & explanations for Classifying lines (HL only)
Math AA exam skills
Paper structures, command terms & tips
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