27 exam-style questions on Vector equation of a line (2D/3D) — topic 3.14 of Geometry and Trigonometry in the IB Mathematics: Analysis and Approaches syllabus — with answers and explanations. Samples below; the full set is free with an account.
A line has vector equation r = (2, 1, 3) + λ(1, −1, 2). Which of the following points lies on the line?
Answer: (4, −1, 7)
With λ = 2: (2 + 2, 1 − 2, 3 + 4) = (4, −1, 7). Each other point matches at most two coordinates for a single value of λ.
Determine a direction vector for the line passing through the points P(2, 1, 0) and Q(5, −1, 4).
Answer: (3, −2, 4)
A direction vector is PQ = Q − P = (5 − 2, −1 − 1, 4 − 0) = (3, −2, 4). Adding the coordinates instead of subtracting gives the trap (7, 0, 4).
A line has vector equation r = (1, 2, 3) + λ(2, 1, −1). Determine its Cartesian equations.
Answer: (x − 1)/2 = y − 2 = (z − 3)/(−1)
Solve each parametric equation for λ: λ = (x − 1)/2 = (y − 2)/1 = (z − 3)/(−1). Watch the signs: the point coordinates are subtracted and the direction components divide.
The lines r = (1, 0) + s(1, 2) and r = (4, 3) + t(−1, 1) intersect. Determine the point of intersection.
Answer: (3, 4)
Equating components: 1 + s = 4 − t and 2s = 3 + t give s = 2, t = 1, so the point is (1 + 2, 0 + 4) = (3, 4).
Consider the lines L₁: r = (1, 1, 1) + s(2, −1, 3) and L₂: r = (0, 3, 2) + t(−4, 2, −6). Determine the relationship between L₁ and L₂.
Answer: They are parallel and distinct
(−4, 2, −6) = −2(2, −1, 3), so the lines are parallel; since (1, 1, 1) does not satisfy the equation of L₂, they are distinct parallel lines.
Consider the lines L₁: r = (1, 0, 0) + s(1, 1, 0) and L₂: r = (0, 0, 1) + t(0, 1, −1). Determine the relationship between L₁ and L₂.
Answer: Skew
The direction vectors are not parallel, and solving 1 + s = 0, s = t, 0 = 1 − t gives t = −1 from the y-equation but t = 1 from the z-equation — no consistent solution, so the lines are skew.
That was 6 of 27 Vector equation of a line (2D/3D) questions.
Practise all 27 freeA student writes the vector equation of the line through A(1, 2) and B(4, 7) as r = (1, 2) + λ(4, 7).
See the correction →27 Vector equation of a line (2D/3D) questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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