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All 8 Flashcards — Vector equation of a line
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Question
What is the vector equation of a line?
Answer
r = a + λd, where a is a point on the line, d is a direction vector, and λ is any real number.
Question
In r = a + λd, what are a and d?
Answer
a = position vector of a known point on the line; d = a direction vector (the line is parallel to it).
Question
How do you find the direction vector through two points A and B?
Answer
d = AB = b − a (subtract the start point's coordinates from the end point's).
Question
Is the vector equation of a line unique?
Answer
No — any point on the line can be a, and any non-zero multiple of d works as the direction.
Question
Line through A(2,1,5) and B(4,5,3): a direction vector?
Answer
d = B − A = (2, 4, −2) (or any multiple, e.g. (1, 2, −1)).
Question
How do you get the parametric form from r = a + λd?
Answer
Write each coordinate on its own line: x = a₁ + λd₁, y = a₂ + λd₂, z = a₃ + λd₃, all sharing λ.
Question
If a direction component is 0, what happens to that coordinate?
Answer
It stays constant — e.g. d = (3, 0, −1) gives y = constant, since y = a₂ + 0·λ.
Question
Two vector equations describe the same line when…
Answer
their directions are parallel (multiples of each other) AND a point of one fits the other.
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Full study notes for Vector equation of a line
Topic 3.14 hub
Vector lines (HL only)
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