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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 B

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.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The points C(−3, 4) and D(5, −2) 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 (−1, 6, 0) and has direction vector \(\begin{pmatrix}4\\-3\\2\end{pmatrix}\).

    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 P(2, 3, −4) and Q(4, −1, −2). Hence determine whether R(8, −9, 2) lies on the line.

  4. A line has vector equation \(\mathbf{r}=\begin{pmatrix}5\\-2\\1\end{pmatrix}+\lambda\begin{pmatrix}-3\\2\\4\end{pmatrix}\). Show that the point E(−4, 4, 12) does not lie on the line.

In context

  1. A zip wire is stretched in a straight line from the top of a tower T(−20, 10, 45) to a post U(40, −20, 15), and the line carries on beyond the post. Coordinates are in metres, with the third value the height.

    1. (a) Find a vector equation of the line.
    2. (b) Determine whether this line passes through a hook at (100, −50, 10).

  2. A spotlight is fixed at S(−3, 4, 1) and points in the direction \(\begin{pmatrix}2\\-1\\3\end{pmatrix}\). Coordinates are in metres.

    1. (a) Write down a vector equation of the beam.
    2. (b) Determine whether the beam passes through the stage mark at (5, 0, 12).

  3. A cable runs from the top of a mast M(0, 0, 50) to an anchor N(80, −60, 0) on the ground. Coordinates are in metres, with the third value the height.

    1. (a) Find a vector equation of the cable.
    2. (b) Find the coordinates of the point on the cable where the height is 20 m.

Stretch

  1. A line has vector equation \(\mathbf{r}=\begin{pmatrix}-2\\5\\1\end{pmatrix}+\lambda\begin{pmatrix}3\\2\\-1\end{pmatrix}\). The point Q(7, k, −2) lies on the line. Find the value of \(k\).

  2. The points C(−2, 4, 6) and D(2, −2, −2) lie on a line \(l\).

    1. (a) Find a vector equation of \(l\).
    2. (b) The midpoint of CD is N. 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