Exercises for Question Type 5: Finding the value of parameters such that two trajectories or objects would intersect

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Exercises for Question Type 5: Finding the value of parameters such that two trajectories or objects would intersect - IB

Question 1

Consider the lines L_{1} and L_{2} defined by: L_{1} : r = 102 + t 3 k 1 L_{2} : r = 021 + s 14 k - 1 Find all values of k such that the lines intersect.

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Question 2

The question asks for the values of parameters m, t, and s such that two given lines in 3D space intersect.

Given the lines L_{1} : (1, - 1, 2) + t (2, m, 4) and L_{2} : (3, 0, 5) + s (1, 2, m), determine all values of m for which they intersect, and find the corresponding values of t and s .

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Question 3

Find the value of m for which the lines L_{1} and L_{2} intersect, where L_{1} : r = 230 + t 11 m L_{2} : r = 041 + s 312 Also find the parameters t and s at the intersection and the intersection point.

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Question 4

Two particles move along paths P (t) = (t, 2 t + 1, 3 t - 2) and Q (s) = (2 s + 1, 4 s - 3, s) . Determine whether the paths intersect and, if so, find the values of t and s at which they meet and the coordinates of the meeting point.

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Question 5

Determine the value of a for which the lines L_{1} : r = 123 + t 2 a 1 and L_{2} : r = 415 + s 132 intersect. Also find the coordinates of the intersection point.

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Question 6

Find all possible values of m, t, and s such that the lines L_{1} : (1, 2, 3) + t (4, m, 6) and L_{2} : (3, 5, 6) + s (2, 8, m + 1) intersect.

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Question 7

The paths of two drones, D_{1} and D_{2}, are given by D_{1} (t) = (r t, 2, t + 1) and D_{2} (s) = (2 s, 2, s) for t, s \in R . Determine the expressions for t and s in terms of r for which the paths of the drones intersect, and state the necessary restriction on r .

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Question 8

Two drones follow paths defined by the vector equations D_{1} (t) = (5, t, 2 t + 3) and D_{2} (s) = (2 s + 1, 4, s - 1) . Determine whether the drones collide.

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Question 9

Do the trajectories L_{1} (t) = (t^{2}, t, 1) and L_{2} (s) = (1, 2 s, s^{2}) intersect? If so, find the values of t and s and the coordinates of the intersection point.

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Question 10

In the plane, determine the value of k such that the lines L_{1} : (01) + t (2 k) and L_{2} : (10) + s (13) intersect. Find the intersection point in terms of k .

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Question 11

Given the lines L_{1} and L_{2} defined by: L_{1} : r = 2 - 1 3 + t a b c L_{2} : r = 111 + s b c a find a condition on a, b, c such that the lines L_{1} and L_{2} intersect.

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IB Mathematics Applications & Interpretation Question Type 5: Finding the Value of Parameters Such That Two Trajectories or Objects Would Intersect Exercises