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IB Mathematics: Applications and Interpretation HL · AHL 4: Vectors · AHL 4.8

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The vector equation of a line Problem set A

Write each line as \(\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}\). Test a point by solving for λ and checking every component. No calculator needed.

Difficulty: first skills applying them stretch

Skills

  1. The points A(2, 5) and B(8, −1) lie on a line.

    1. (a) Find a vector equation of the line.
    2. (b) Find the point on the line when \(\lambda=\frac{1}{2}\).

  2. A line passes through the point (3, −2, 4) and has direction vector \(\begin{pmatrix}1\\5\\-2\end{pmatrix}\).

    1. (a) Write down a vector equation of the line.
    2. (b) Write down a second vector equation of the same line, using the point on the line where \(\lambda=1\) as the position vector.

  3. Find a vector equation of the line through A(1, −2, 5) and B(3, 4, 1). Hence determine whether C(4, 7, −1) lies on the line.

  4. A line has vector equation \(\mathbf{r}=\begin{pmatrix}0\\4\\-3\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\5\end{pmatrix}\). Show that the point D(6, 1, 10) does not lie on the line.

In context

  1. A drone flies in a straight line through P(0, 0, 30) and then Q(80, 60, 10), and keeps going on the same line. Coordinates are in metres, with the third value the height.

    1. (a) Find a vector equation of the line.
    2. (b) Determine whether the drone reaches a landing pad at (120, 90, 0).

  2. A laser is fixed at L(2, −1, 3) and points in the direction \(\begin{pmatrix}1\\2\\-2\end{pmatrix}\). Coordinates are in metres.

    1. (a) Write down a vector equation of the beam.
    2. (b) Determine whether the beam passes through the target at (7, 9, −7).

  3. A zip wire runs from the top of a tower T(0, 0, 40) to a point G(120, 50, 0) on the ground. Coordinates are in metres, with the third value the height.

    1. (a) Find a vector equation of the wire.
    2. (b) Find the coordinates of the point on the wire where the height is 10 m.

Stretch

  1. A line has vector equation \(\mathbf{r}=\begin{pmatrix}1\\0\\4\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\3\end{pmatrix}\). The point P(k, −2, 10) lies on the line. Find the value of \(k\).

  2. The points A(1, −3, 2) and B(5, 3, −6) lie on a line \(l\).

    1. (a) Find a vector equation of \(l\).
    2. (b) The midpoint of AB is M. Show that it lies on \(l\) and state the value of \(\lambda\) there.
    3. (c) Find the coordinates of the point on \(l\) where \(\lambda=3\).

jmaths.xyz · AI HL · AHL 4.8 The vector equation of a line