Practice Flashcards
What is the vector equation of a line?
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All Flashcards in Topic 3.14
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3.14.18 cards
What is the vector equation of a line?
r = a + λd, where a is a point on the line, d is a direction vector, and λ is any real number.
In r = a + λd, what are a and d?
a = position vector of a known point on the line; d = a direction vector (the line is parallel to it).
How do you find the direction vector through two points A and B?
d = AB = b − a (subtract the start point's coordinates from the end point's).
Is the vector equation of a line unique?
No — any point on the line can be a, and any non-zero multiple of d works as the direction.
Line through A(2,1,5) and B(4,5,3): a direction vector?
d = B − A = (2, 4, −2) (or any multiple, e.g. (1, 2, −1)).
How do you get the parametric form from r = a + λd?
Write each coordinate on its own line: x = a₁ + λd₁, y = a₂ + λd₂, z = a₃ + λd₃, all sharing λ.
If a direction component is 0, what happens to that coordinate?
It stays constant — e.g. d = (3, 0, −1) gives y = constant, since y = a₂ + 0·λ.
Two vector equations describe the same line when…
their directions are parallel (multiples of each other) AND a point of one fits the other.
3.14.28 cards
How do you test whether a point lies on a line r = a + λd?
Solve for λ from one coordinate, then check the SAME λ satisfies every other coordinate. All must agree.
How do you find the point on a line for a given λ?
Substitute that value of λ into r = a + λd and compute each coordinate.
Where does a line cross the x-axis (in 2D)?
Where y = 0: set the y-equation to 0, solve for λ, then substitute back for x.
In 3D, a point is on the z-axis when…
x = 0 AND y = 0 (only the z-coordinate is free).
In the motion model r = a + t·d, what is the speed?
Speed = |d|, the magnitude of the velocity (direction) vector.
Speed of an object with velocity (4, 0, −3)?
√(4² + 0² + (−3)²) = √25 = 5.
What does the parameter t mean in a motion model r = a + t·d?
t is the time; a is the start position (t = 0) and d is the constant velocity.
If a point gives different λ values in different rows, is it on the line?
No — the point is off the line; one λ must satisfy all coordinates simultaneously.
Topic 3.14 study notes
Full notes & explanations for Vector lines (HL only)
Math AA exam skills
Paper structures, command terms & tips
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