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All 8 Flashcards — Angle between two lines
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Question
Formula for the angle between two lines?
Answer
cos θ = |d₁·d₂| / (|d₁| |d₂|), where d₁, d₂ are the direction vectors.
Question
Why does the angle between two lines use only the direction vectors?
Answer
Sliding a line (keeping its direction) doesn't change the angle, so the base points are irrelevant — only the directions matter.
Question
Why the absolute-value bars in cos θ = |d₁·d₂|/(|d₁||d₂|)?
Answer
They keep cos θ positive so you report the ACUTE angle; a negative dot product would otherwise give an obtuse angle.
Question
How do you test whether two lines are perpendicular?
Answer
Show their direction vectors have dot product zero: d₁·d₂ = 0 ⟺ perpendicular.
Question
Angle between directions (1, −1, 2) and (2, 1, 1)?
Answer
Dot = 3, |d₁| = |d₂| = √6, cos θ = 3/6 = ½ ⇒ θ = 60°.
Question
If d₁·d₂ is negative, what does that tell you?
Answer
The arrows make an obtuse angle; take the modulus to get the acute angle between the lines.
Question
Do the lengths of the direction vectors change the angle?
Answer
No — the formula divides by both magnitudes, so any scaling of a direction cancels out.
Question
What is the denominator in the angle formula?
Answer
The PRODUCT of the magnitudes |d₁| × |d₂| (not their sum).
Read the notes
Full study notes for Angle between two lines
Topic 3.15 hub
Classifying lines (HL only)
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Math AA exam skills
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