Vector equation of a line
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Flip to reveal answersWhat does the vector equation of a line r = a + λd mean?
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Question
What does the vector equation of a line r = a + λd mean?
Answer
a is the position vector of a point ON the line, d is the direction the line points, and λ is a number that slides you along it (λ = 0 gives a, λ = 1 gives a + d).
Question
In r = a + λd, which part is the direction and which is a point on the line?
Answer
d (the vector multiplied by λ) is the direction; a (the constant part) is a point on the line.
Question
How do you find the direction vector of a line through points A and B?
Answer
d = B − A (finish minus start). Any scalar multiple of it is also a valid direction.
Question
Write the vector equation of the line through points A and B.
Answer
r = A + λ(B − A) — start at A, walk in the direction B − A.
Question
How do you find an object's position at a given value of λ?
Answer
Substitute that number for λ and add component by component: r = a + λd.
Question
How do you test whether a point P lies on the line r = a + λd?
Answer
Set a + λd = P, solve ONE component for λ, then check the SAME λ works in every other component. One λ fits all → on the line; otherwise → off it.
Question
Does a vector line have only one possible equation?
Answer
No — different start points a (any point on the line) and any scalar multiple of d describe the same line.
Question
Where on the line is the midpoint of A and B, in terms of λ?
Answer
At λ = ½ in r = A + λ(B − A): a half-step of the direction from A.
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Topic 3.11 hub
Vector equation of a line (HL only)
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