IB Mathematics Analysis and Approaches Topic AHL…

Two lines in three-dimensional space are given by $L_1: \mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad L_2: \mathbf{r}=\begin{pmatrix}5\\0\\1\end{pmatrix}+\mu\begin{pmatrix}1\\-2\\2\end{pmatrix},\qquad \lambda,\mu\in\mathbb{R}.$

(a) Write both lines in parametric and Cartesian form.

(b) whether the lines are skew, parallel, coincident, or intersecting. If they intersect, find the point of intersection.

(d) Let $S = (2, 0, 5)$ . Find the shortest distance from $S$ to the line $L_1$ , and the coordinates of the foot $F$ of the perpendicular from $S$ to $L_1$ .

(e) Find all points $Q$ on $L_2$ such that the vector $\overrightarrow{PQ}$ makes an angle of $45^\circ$ with the direction of $L_1$ , where $P$ is the intersection point from part (b). Give exact parameter values $\mu$ and coordinates of $Q$ if they exist.

The lines $L_{1}$ and $L_{2}$ have vector equations

$L_{1}:\ \vec{r} = \begin{pmatrix}-1\\1\\0\end{pmatrix} + \lambda\begin{pmatrix}1\\p\\0\end{pmatrix}, \qquad L_{2}:\ \vec{r} = \begin{pmatrix}0\\0\\1\end{pmatrix} + \mu\begin{pmatrix}0\\1\\p\end{pmatrix},$

where $\lambda,\,\mu \in \mathbb{R}$ and $p$ is a real number.

(a) Show that $L_{1}$ and $L_{2}$ have no point in common, whatever the value of $p$ .

(b) Hence show that the two lines are skew for every real value of $p$ .

Line $L_1$ : $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$ and plane $\pi$ : $2x - y + z = 5$ .

(a) Find the point of intersection of the line $L_1$ and the plane $\pi$ .

(b) if the line $L_1$ is parallel, perpendicular, or neither to the normal of the plane $\pi$ .

IB Mathematics Analysis and Approaches AHL 3.15—classification of Lines Questionbank

IB Mathematics Analysis and Approaches AHL 3.15—classification of Lines Questionbank

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IB Mathematics Analysis and Approaches AHL 3.15—classification of Lines Questionbank

IB Mathematics Analysis and Approaches AHL 3.15—classification of Lines Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 3.15—classification of Lines Questionbank

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