3.17 Vector equations of planes 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 $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}$.

Hence find the Cartesian equation of the plane $\mathrm{\Pi }$ that contains A, B and C.

Show that D lies in $\mathrm{\Pi }$.

The quadrilateral ABCD is convex. Find its area.

Find $\overrightarrow{\mathrm{A}\mathrm{C}}\times \overrightarrow{\mathrm{B}\mathrm{D}}$, and verify that ${\displaystyle \frac{1}{2}}\mid \overrightarrow{\mathrm{A}\mathrm{C}}\times \overrightarrow{\mathrm{B}\mathrm{D}}\mid$ is equal to the area of ABCD.

Consider any convex quadrilateral ABCD whose vertices lie in one plane. In such a quadrilateral, the vectors $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}$ and $\overrightarrow{\mathrm{A}\mathrm{C}}\times \overrightarrow{\mathrm{A}\mathrm{D}}$ have the same direction.

Show that the area of ABCD is ${\displaystyle \frac{1}{2}}\mid \overrightarrow{\mathrm{A}\mathrm{C}}\times \overrightarrow{\mathrm{B}\mathrm{D}}\mid$.

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 $\overrightarrow{\mathrm{O}\mathrm{P}}\times \overrightarrow{\mathrm{O}\mathrm{R}}$.

Find the area of the panel.

Find the acute angle between the panel and the horizontal ground.

At a particular time of day, the rays of the sun travel in the direction $\left(\begin{array}{c}1\\ -2\\ -6\end{array}\right)$. The panel works best when the angle between the sun's rays and the normal to the panel is small.

Find the acute angle between the sun's rays and the normal to the panel at this time.

When the sun is directly overhead, the shadow of the panel on the ground is the parallelogram with adjacent sides $\left(\begin{array}{c}2.4\\ 0\\ 0\end{array}\right)$ and $\left(\begin{array}{c}0.3\\ 1.6\\ 0\end{array}\right)$.

Find the area of the shadow.

Show that the area of the shadow is equal to the area of the panel multiplied by the cosine of the angle found in part (c).

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 $\phi$.