Vector product: notes and practice questions

Two spacecraft, S1 and S2, travel along straight paths, represented by the lines $L_1$ and $L_2$ respectively. The paths of the spacecraft intersect at a docking station D. A probe is located at a point P on the path of $L_2$. This is shown in the following diagram.

The direction vector of $L_1$ is $\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$. The vector $\vec{\text{DP}}$ is given by $\begin{pmatrix} k \\ 4 \\ 0 \end{pmatrix}$, where $k \ge 0$.

The acute angle between the paths $L_1$ and $L_2$ is $\theta$, where $\cos\theta = \frac{1}{3}$.

(a) Show that $2k+4 = \sqrt{k^2+16}$.

(b) Find the value of $k$.

(c) Hence, find the shortest distance from the probe at P to the path $L_1$.

The paths $L_1$ and $L_2$ lie on a plane, $\Pi$.

(d) Find a vector normal to the plane $\Pi$.

A satellite dish is modelled as a right circular cone with its vertex at V. The base of the cone lies in the plane $\Pi$ and is centred at P. The path $L_1$ is tangent to the circular base of the cone. The volume of the cone is $\frac{128\pi\sqrt{2}}{9}$ cubic units. The position vector of P is $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.

(e) Find the two possible position vectors for V.

The paths of two submarines, A and B, are described by the vector equations below, where $s, t \in \mathbb{R}$ are time parameters in hours, and the coordinates are in kilometres.

$L_A: \boldsymbol{r} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$

$L_B: \boldsymbol{r} = \begin{pmatrix} 11 \\ 2 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$

(a) Show that the paths of the two submarines do not cross.

(b) Find the shortest distance between the paths of the two submarines.

(a) The line $L_1$ passes through the point Q(2, 0, 5) and has a direction vector $\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$.

Write down a vector equation for $L_1$.

(b) A second line, $L_2$, passes through the points C(3, 1, 0) and D(4, 3, -2).

Find a vector equation for $L_2$.

(c) Show that $L_1$ and $L_2$ are skew.

(d) Find $\vec{\text{QM}} \cdot \vec{\text{CD}}$ in terms of $\mu$, where M is a general point on $L_2$.

(e) Hence, find the coordinates of the point M on $L_2$ that is closest to Q.

(f) The origin is denoted by O(0, 0, 0). Find the equation of the plane that contains the points O, Q and the point M found in part (e). Give your answer in the form $ax + by + cz = d$, where $a, b, c, d \in \mathbb{Z}$.

A team of architects is designing a new building and needs to define a support beam's orientation. The beam must be perpendicular to two existing structural walls, Wall A and Wall B. The equations of the planes representing these walls are given by:

Wall A ($\Pi_A$): $x + 2y - z = 7$

Wall B ($\Pi_B$): $2x - y + 3z = 1$

Find a Cartesian equation of the plane ($\Pi_C$) that represents the orientation of the support beam, given that it passes through the origin (0, 0, 0).

Find the coordinates of the point where Wall A, Wall B, and the support beam's plane ($\Pi_C$) intersect.

Consider the points given by the coordinates $P(2, 1, 0)$, $Q(0, 3, 1)$, $R(1, 0, 4)$.

Find the vector $\vec{PQ} \times \vec{PR}$.

Hence, find the exact area of triangle PQR.

Show that the Cartesian equation of the plane $\Pi_1$, which contains the triangle PQR, is $9x + 7y + 4z = 25$.

A second plane is given by the equation $\Pi_2: x + y - z = 1$. Find a vector equation for the line of intersection of the planes $\Pi_1$ and $\Pi_2$.

A landscape architect is designing a triangular shade sail for a patio. The vertices of the sail are defined by points A, B, and C in a 3D coordinate system, where the z-axis represents height.

The coordinates of the vertices are A$(0, k, 1)$, B$(2, 1, 0)$ and C$(k, 0, 3)$, where $k$ is a positive constant representing a design parameter.

(a) Show that the vector product $\vec{AB} \times \vec{AC}$ is given by $\begin{pmatrix} 2-3k \\ -k-4 \\ k^2-3k \end{pmatrix}$.

(b) The architect wants to minimize the tension in the sail, which is proportional to the magnitude of the vector product of two adjacent sides. Find the smallest possible value of $|\vec{AB} \times \vec{AC}|$.

(c) Calculate the smallest possible area of the shade sail.

Two drones, Drone Alpha and Drone Beta, are flying in a 3D space. At a particular instant, Drone Alpha passes through point $A(1, 2, 3)$ and then point $B(5, 0, 1)$.

(a) Find a vector equation of the line representing Drone Alpha's path.

At the same instant, Drone Beta passes through point $C(2, -1, 4)$ and then point $D(0, 3, 2)$.

(b) Find a vector equation of the line representing Drone Beta's path.

(c) Hence, or otherwise, find the shortest distance between the paths of Drone Alpha and Drone Beta.

Consider the non-zero vectors $\vec{u}$ and $\vec{v}$. Let $\theta$ be the angle between $\vec{u}$ and $\vec{v}$.

Using the definitions of $\vec{u} \cdot \vec{v}$ and $\vec{u} \times \vec{v}$ in terms of $|\vec{u}|$, $|\vec{v}|$ and $\theta$, show that $(\vec{u} \cdot \vec{v})^2 + |\vec{u} \times \vec{v}|^2 = |\vec{u}|^2|\vec{v}|^2$.

A triangle PQR has vertices P(1, 0, 1), Q($a, b$, 2) and R(4, 1, 1), where $a, b \in \mathbb{Q}$.

The vectors $\vec{u}$ and $\vec{v}$ are defined as $\vec{u} = \vec{PQ}$ and $\vec{v} = \vec{PR}$.

It is given that $\vec{u} \cdot \vec{v} = 4$ and the area of triangle PQR is $\frac{\sqrt{14}}{2}$ square units.

Find the value of $|\vec{u} \times \vec{v}|$.

Hence, or otherwise, find the value of $|\vec{u}|$.

Hence, or otherwise, find the possible values of $a$ and the corresponding values of $b$.

Consider a new point S, the vector $\vec{w}$ is defined as $\vec{w} = \vec{RS}$.

It is given that $\vec{u} \cdot \vec{w} = 0$ and $\vec{v} \cdot \vec{w} = 0$, and the area of triangle PRS is 10 square units.

Assuming that $a = 2$, find the possible vectors for $\vec{w}$.

Two vectors are given by $\vec{u} = \mathbf{i} + p\mathbf{j} + (p+1)\mathbf{k}$ and $\vec{v} = (p-2)\mathbf{i} - \mathbf{j} + 2\mathbf{k}$, where $p \in \mathbb{R}$.

(a) Find the value of $p$ for which the vectors $\vec{u}$ and $\vec{v}$ are orthogonal.

(b) For this value of $p$, find the Cartesian equation of the plane that contains the vectors $\vec{u}$ and $\vec{v}$ and passes through the point $(3, -1, 0)$.

Let P(1, 0, 1), Q(1, 2, 0), and R(k+1, 1, -1) be three points in $\mathbb{R}^3$, where k > 0.

Let $Π$ be the plane containing the points P, Q, and R.

(a) Find a Cartesian equation for the plane $Π$ in terms of k.

(b) Let N be the midpoint of the line segment [PR]. A line L passes through N and is perpendicular to the plane $Π$. Find a vector equation for the line L in terms of k.

(c) Let $L'$ be the line defined by the equations $y=x, z=1$. Show that the line L does not intersect the line $L'$ for any k > 0.

A structural engineer is designing a framework and needs to define the orientation of certain surfaces. Consider two existing planar surfaces, $\Pi_1$ and $\Pi_2$, with the following Cartesian equations:

$\Pi_1\text{: } 2x + y - z = 5$

$\Pi_2\text{: } x - 3y + 2z = -1$

Find a Cartesian equation of a third planar surface, $\Pi_3$, which is perpendicular to both $\Pi_1$ and $\Pi_2$, and passes through the point $(1, 2, 3)$.

Determine the coordinates of the point where $\Pi_1$, $\Pi_2$, and $\Pi_3$ intersect.