AHL 3.11—Vector equation of a line in 2d and 3d Questionbank

A line $L$ has the vector equation

$\vec{r}=\begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}+\lambda\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$

A line $L$ is defined by the vector equation

A line passes through the point $P(4,-1,2)$ and is parallel to the vector $\begin{pmatrix} 2 \\ 0 \\ -3 \end{pmatrix}$.

The line $L_{1}$ is given by $\vec{r}=\begin{pmatrix}1 \\ 3 \\ 2\end{pmatrix}+\lambda\begin{pmatrix}2 \\ -1 \\ 1\end{pmatrix}$ and point $A=(5,1,3)$.

A line in 3D passes through the points $P(0, 1, -2)$ and $Q(2, 3, 0)$.

The position vectors of points A and B are $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $7\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ respectively.

At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow.

The path of the ball is modelled by the equation $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 0 \end{pmatrix} + t \begin{pmatrix} u_x \\ u_y - 5t \end{pmatrix}$ where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacement from the ground, both measured in metres, and $t$ is the time, in seconds, since the ball was launched.

• $u_x$ is the horizontal component of the initial velocity
• $u_y$ is the vertical component of the initial velocity.

In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$.

An archer releases an arrow from the point $(0, 2)$. The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed $60 \text{ m s}^{-1}$ and an angle of elevation of $10^\circ$.

Consider a vector line in 2D represented by the vector equation $\mathbf{r} = \mathbf{a} + t\mathbf{b}$, where $\mathbf{a}$ is a position vector and $\mathbf{b}$ is a direction vector.

Two lines $L_1$ and $L_2$ are given by the following equations, where $p \in \mathbb{R}$.

$L_1: \mathbf{r} = \begin{pmatrix} 2 \\ p+9 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} p \\ 2p \\ 4 \end{pmatrix}$

$L_2: \mathbf{r} = \begin{pmatrix} 14 \\ 7 \\ p+12 \end{pmatrix} + \mu \begin{pmatrix} p+4 \\ 4 \\ -7 \end{pmatrix}$

It is known that $L_1$ and $L_2$ are perpendicular.

At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow. The path of the ball is modelled by the equation $\binom{x}{y} = \binom{5}{0} + t \binom{u_x}{u_y - 5t}$ where $x$ is the horizontal displacement from the archer and $y$ is the vertical displacement from the ground, both measured in metres, and $t$ is the time, in seconds, since the ball was launched.

• $u_x$ is the horizontal component of the initial velocity
• $u_y$ is the vertical component of the initial velocity. In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that $u_x = 8$ and $u_y = 10$