Topic 3.11: Vector equation of a line (HL only) Questions

On the line r = (1, 3) + λ(2, −2), the point reached at λ = 2 is:

In the vector equation r = a + λd of a line, the vector a represents:

A cyclist's position (km) is given by r = (3, 1) + λ(4, 2), where λ is in hours. Find the position after 2 hours and state the direction of travel.

A line passes through C(2, 5) and D(8, 1). A direction vector for the line is:

Which statement about the line r = (0, 0) + λ(2, 6) is correct?

A straight zip-line runs from tower A(2, 10, 30) to tower B(14, 4, 6) (metres, third component = height). Find a vector equation of the zip-line.

Find a vector equation of the line through $A$ and $B$.

A line has equation r = (5, −2) + λ(−1, 3). Determine whether the point (2, 7) lies on the line.

To test whether the point P lies on the line r = a + λd, you should:

Two delivery vans set off. Van A's position is r = (0, 0) + λ(3, 4) and van B's is r = (12, 1) + μ(−1, 2) (km). The vans collide if their paths cross at the same parameter point. Find where van A is when λ = 3 and state its distance from the start.

Show that the point $(7,0)$ lies on this line.

A search-and-rescue helicopter flies along r = (−2, 5, 0) + λ(4, −1, 2) (km, third component = altitude). A stranded climber is at the point (10, 2, 6). Determine whether the flight path passes directly over (through) the climber's position.

A line passes through P(1, 0, 2) and Q(4, 6, −1) (metres). (a) Find a vector equation. (b) Find the point on the line where λ = 2 and interpret how far along PQ it is.