3.16 The vector product questions for IB Maths AA

Show that $L_1$ and $L_2$ are skew.

Find $\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)\times \left(\begin{array}{c}0\\ 1\\ 2\end{array}\right)$.

The point P lies on $L_1$ and the point Q lies on $L_2$ such that $[\mathrm{P}\mathrm{Q}]$ is perpendicular to both $L_1$ and $L_2$.

Explain why $\overrightarrow{\mathrm{P}\mathrm{Q}}=k\left(\begin{array}{c}2\\ -2\\ 1\end{array}\right)$ for some $k\in \mathbb{R}$.

Find the coordinates of P and the coordinates of Q.

Hence find the shortest distance between $L_1$ and $L_2$.

The plane $\mathrm{\Pi }$ contains $L_1$ and is parallel to $L_2$.

Find the Cartesian equation of $\mathrm{\Pi }$.

Find the coordinates of A.

The line $L_2$ intersects $L_1$ at the point B and intersects $L_3$ at the point C.

Find the coordinates of B and the coordinates of C.

Find $\mathrm{B}\widehat{\mathrm{A}}\mathrm{C}$.

Find the area of triangle ABC.

Hence find the shortest distance from C to the line $L_1$.

Consider the line $L_2$.

Write down the coordinates of the point on $L_2$ for which $\lambda =3$.

Verify that this point lies on $S$.

Show that every point of $L_a$ lies on $S$.

Show that $L_1$ and $L_2$ are skew.

Show that, for $a\mathrm{\ne }c$, the lines $L_a$ and $L_c$ are skew.

Show that $L_a$ and $M_b$ intersect, and find the coordinates of their point of intersection in terms of $a$ and $b$.

The point $\mathrm{P}(p,q,pq)$ lies on $S$.

Write down the value of $a$ and the value of $b$ for which both $L_a$ and $M_b$ pass through P.

Let $\theta$ be the acute angle between $L_p$ and $M_q$.

Show that $\cos \theta ={\displaystyle \frac{\mathrm{\mid }pq\mathrm{\mid }}{\sqrt{(1+p^2)(1+q^2)}}}$.

Hence show that $L_p$ and $M_q$ are perpendicular if and only if P lies on the $x$-axis or on the $y$-axis.

The plane $\mathrm{\Pi }$ contains both $L_p$ and $M_q$.

Show that the Cartesian equation of $\mathrm{\Pi }$ is $qx+py-z=pq$.

By substituting $z=xy$ into the equation of $\mathrm{\Pi }$, show that the only points of $S$ that lie in $\mathrm{\Pi }$ are the points of $L_p$ and $M_q$.

Find the value of $k$ for which $L_1$ and $L_2$ are parallel.

For this value of $k$, show that $L_1$ and $L_2$ are not coincident.

Find the value of $k$ for which $L_1$ and $L_2$ intersect, and find the coordinates of the point of intersection.

State the relationship between $L_1$ and $L_2$ when $k\mathrm{\ne }0$ and $k\mathrm{\ne }4$.

For the value of $k$ found in part (c), the plane ${\mathrm{\Pi }}_1$ contains both $L_1$ and $L_2$.

Find the Cartesian equation of ${\mathrm{\Pi }}_1$.

For the value of $k$ found in part (a), the plane ${\mathrm{\Pi }}_2$ contains both $L_1$ and $L_2$.

Find the Cartesian equation of ${\mathrm{\Pi }}_2$.

Find $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}$.

Hence find the area of the sail.

Find $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}$.

Find the exact area of triangle ABC.

The points ${\mathrm{A}}^{\mathrm{\prime }}$, ${\mathrm{B}}^{\mathrm{\prime }}$ and ${\mathrm{C}}^{\mathrm{\prime }}$ are the projections of A, B and C onto the $xy$-plane.

Use a vector product to find the area of triangle ${\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\mathrm{C}}^{\mathrm{\prime }}$.

Now consider any triangle ABC, with $\overrightarrow{\mathrm{A}\mathrm{B}}=\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\end{array}\right)$, $\overrightarrow{\mathrm{A}\mathrm{C}}=\left(\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right)$ and $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}=\left(\begin{array}{c}{n}_1\\ n_2\\ n_3\end{array}\right)$.

Show that the area of the projection of triangle ABC onto the $xy$-plane is ${\displaystyle \frac{1}{2}}\mathrm{\mid }{n}_3\mathrm{\mid }$.

Write down, in terms of $n_1$, $n_2$ and $n_3$, the area of the projection of triangle ABC onto the $yz$-plane.

Write down, in terms of $n_1$, $n_2$ and $n_3$, the area of the projection of triangle ABC onto the $xz$-plane.

Let $T$ be the area of triangle ABC, and let $X$, $Y$ and $Z$ be the areas of its projections onto the $yz$-, $xz$- and $xy$-planes respectively.

Show that $T^2=X^2+Y^2+Z^2$.

Verify this result for the triangle in part (a).

The tetrahedron OPQR has vertices $\mathrm{O}(0,0,0)$, $\mathrm{P}(a,0,0)$, $\mathrm{Q}(0,b,0)$ and $\mathrm{R}(0,0,c)$, where $a,b,c>0$.

Find $\overrightarrow{\mathrm{P}\mathrm{Q}}\times \overrightarrow{\mathrm{P}\mathrm{R}}$.

Hence show that the areas of the four faces of the tetrahedron satisfy

Suggest why this result is sometimes described as a three-dimensional version of Pythagoras' theorem.

Consider the case $a=2$ and $b=3$.

Find the value of $c$ for which the area of face PQR is 7.

For the tetrahedron in part (k), let $\phi$ be the acute angle between the face PQR and the face OPQ.

Show that $\cos \phi ={\displaystyle \frac{\text{area of OPQ}}{\text{area of PQR}}}$, and hence find $\phi$.