Vector Planes (DP IB Analysis & Approaches (AA): HL): Exam Questions

The points A(2, 1, 0), B(-1, 4, 1) and C(1, 0, 3) lie on a plane $\Pi$.

Find an equation for $\Pi$ in the form $ax+by+cz=d$ where $a,b,c,d\in \mathrm{\mathbb{Z}}.$

Determine whether the point D(-2, 2, 5) lies on $\Pi$.

The plane $\Pi$ has equation $r·\left(\begin{array}{c}4\\ -3\\ 1\end{array}\right)=8$.

The line $L$ has equation $r=\left(\begin{array}{c}2\\ -1\\ 5\end{array}\right)+s\left(\begin{array}{c}1\\ 2\\ 4\end{array}\right)$. 

The plane $\Pi$ and the line $L$ intersect at the point X.

Find the coordinates of X.

Find the acute angle, in degrees, between the line $L$ and the plane $\Pi$.

The point P(1,-3, 1) lies on the line $L$.

Find the exact value of PX.

Hence find the shortest distance between the point P and the plane $\Pi$.

Find the acute angle, in radians, between the two planes ${\Pi }_1$ and ${\Pi }_2$ which can be defined by the equations:

${\Pi }_1:5x-2y+z=19$

${\Pi }_2:\mathrm{r}·\left(\begin{array}{c}3\\ 5\\ -2\end{array}\right)=20$.

The line L given by the Cartesian equation $\frac{x-1}{2}=\frac{3-y}{3}=z+2$ lies on the plane $\Pi .$The point P(4, 0, -3)  also lies on $\Pi .$

Show that the vectors $\left(\begin{array}{c}2\\ -3\\ 1\end{array}\right)$ and $\left(\begin{array}{c}1\\ 0\\ -2\end{array}\right)$ are parallel to $\Pi .$

Hence find the Cartesian equation of $\Pi$ .

Consider the plane $\Pi$ defined by the Cartesian equation $2x-5y+3z=19$ and the line $L_1$ defined by the vector equation $\mathrm{r}=\left(\begin{array}{c}7\\ -4\\ 9\end{array}\right)+\lambda \left(\begin{array}{c}4\\ 1\\ -1\end{array}\right)$.

Show that the line $L_1$ is parallel to the plane $\Pi$ but does not lie in the plane.

The line $L_2$  is perpendicular to the plane $\Pi$ and passes through the point P(7, -4, 9) .

Find a vector equation of the line $L_2$.

Find the coordinates of the point where the line  $L_2$and the plane$\Pi$intersect.

Hence find the shortest distance between the line $L_1$ and the plane $\Pi$.

Consider the two planes defined by the Cartesian equations:

${\Pi }_1:2x+y+2z=8$

${\Pi }_2:3x-y-2z=7.$ 

The line $L$ is the intersection of the planes ${\Pi }_1$ and ${\Pi }_2$.

Show that the line $L$ is parallel to the vector $\left(\begin{array}{c}0\\ 2\\ -1\end{array}\right)$.

The point P$\left(a,0,b\right)$ lies on both planes.

(i) Find the values of $a$ and $b$.

(ii) Hence write down a vector equation of the line $L$.

A third plane ${\Pi }_3$ has the Cartesian equation $2x-3y+z=14$.

Use algebra to show that the three planes intersect at a unique point Q and find the coordinates of Q.

Consider the three planes with Cartesian equations:

${\Pi }_1:2x+3y+kz=11$

${\Pi }_2:3x+y-z=-8$

${\Pi }_3:x-5y+2z=15$

where $k$ is a real constant. 

In the case when the three planes do not intersect at a unique point, find the value of $k$  and state the geometrical relationship between the three planes.

In the case when $k=0$ find the coordinates of the point of intersection between the three planes.

Two parallel planes are defined by the equations:

 ${\Pi }_1:\mathrm{r}·\left(\begin{array}{c}7\\ -4\\ a\end{array}\right)=113,a\in \mathrm{ℝ},$

${\Pi }_2:\mathrm{r}=\left(\begin{array}{c}11\\ -3\\ 5\end{array}\right)+\lambda \left(\begin{array}{c}b\\ 4\\ -1\end{array}\right)+\mu \left(\begin{array}{c}7\\ -2\\ -3\end{array}\right),b\in \mathrm{ℝ}.$

Show that $a=19$ and find the value of $b$.

Write down a vector equation of the line $L$ that is perpendicular to both planes and goes through the point P(11, -3, 5).

Find the coordinates of the point where the line $L$ intersects the plane ${\Pi }_1.$

Hence find the shortest distance between the two planes and ${\Pi }_2$.

The plane $\Pi$ has the vector equation $\mathrm{r}=\left(\begin{array}{c}6\\ -19\\ -6\end{array}\right)+\lambda \left(\begin{array}{c}7\\ -3\\ 1\end{array}\right)+\mu \left(\begin{array}{c}-2\\ 8\\ -1\end{array}\right).$

Find a vector that is perpendicular to the plane $\Pi$ .

Q is the point on the plane $\Pi$  that is closest to the point P(4, 0, -3). Find the coordinates of the point Q.

Hence find the reflection of the point P in the plane $\Pi .$

Two planes are defined by the Cartesian equations:

${\Pi }_1:x-2y+3z=11$

${\Pi }_2:3x+4y-z=3.$

Find the acute angle, in radians, between ${\Pi }_1$and ${\Pi }_2$.

A third plane ${\Pi }_3$ is defined by the equation  $5x+ky+z=13$where $k\in \mathrm{ℝ}$.

The plane ${\Pi }_3$ is perpendicular to the plane ${\Pi }_1$. Find the value of $k$.

(i) Describe the geometrical configuration of the three planes.

(ii) Find the acute angle, in radians, between ${\Pi }_2$ and ${\Pi }_3$.

Determine whether the points A(1, -1, 8) , B(0, 10, 15) , C (-2, -6. 10) and D(3, -5, 3) can lie in the same plane.

The plane $\Pi$ has vector equation $\mathrm{r}=\left(\begin{array}{c}-1\\ 5\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}3\\ -2\\ -2\end{array}\right)+\mu \left(\begin{array}{c}4\\ 1\\ 5\end{array}\right)$ 

The line $L$ has vector equation $\mathrm{r}=\left(\begin{array}{c}0\\ 4\\ 5\end{array}\right)+s\left(\begin{array}{c}1\\ 1\\ -3\end{array}\right)$ 

The plane $\Pi$ and the line $L$ intersect at the point $\mathrm{X}$. 

Find the coordinates of $\mathrm{X}$.

Find the acute angle, in degrees, between the line $L$ and the plane $\Pi$.

The point P(2, 6, -1) lies on the line $L$.

Find the shortest distance between the point P and the plane $\Pi$. Fully justify your answer.

Find the acute angle, in radians, between the two planes ${\Pi }_1$and ${\Pi }_2$ which can be defined by the equations:

${\Pi }_1:7x+3y-2z=84,$

${\Pi }_2:\mathrm{r}=\left(\begin{array}{c}11\\ -7\\ 9\end{array}\right)+\lambda \left(\begin{array}{c}-2\\ 5\\ 0\end{array}\right)+\mu \left(\begin{array}{c}1\\ 6\\ -4\end{array}\right)$.

The plane ${\Pi }_1$is defined by the equation $x-2y-2z+15=0$and the line $L$ is defined by the vector equation  $\mathrm{r}=\left(\begin{array}{c}-5\\ 1\\ 4\end{array}\right)+\lambda \left(\begin{array}{c}4\\ -3\\ 5\end{array}\right)$.

Show that the line $L$ lies on the plane ${\Pi }_1.$

The plane ${\Pi }_2$ is defined by the equation $\mathrm{r}=\left(\begin{array}{c}3\\ 1\\ 4\end{array}\right)+s\left(\begin{array}{c}12\\ -1\\ 7\end{array}\right)+t\left(\begin{array}{c}2\\ 5\\ -4\end{array}\right)$,

Show that the plane  ${\Pi }_2$ is parallel to the plane ${\Pi }_1$.

Find a vector equation of the line that is perpendicular to both planes and passes through the point P(3, 1, 4).

Hence find the shortest distance between  ${\Pi }_1$ and ${\Pi }_2$.

The plane $\Pi$ has the Cartesian equation $x+4y+2z+25=0$. 

The line $L$has the Cartesian equation $\frac{3-x}{2}=k\left(y+2\right)=z+\frac{1}{5},$ where $k\in \mathrm{ℝ}.$

Show that the $L$ is not parallel to the plane $\Pi$.

Given that the acute angle between the line $L$ and the plane $\Pi$ is 60°, find the possible values of $k$.

Consider the two planes defined by the Cartesian equations:

${\Pi }_1:3x-5y+2z=9$

${\Pi }_2:4x+2y-z=13.$ 

The line $L$ is the intersection of the planes ${\Pi }_1\mathrm{and}{\Pi }_2.$ 

Find a vector equation of the line $L$. Give your answer in the form $\mathrm{r}=\left(\begin{array}{c}1\\ a\\ b\end{array}\right)+\lambda \left(\begin{array}{c}c\\ d\\ e\end{array}\right)$where $a,b,c,d,e\in \mathrm{\mathbb{Z}}$.

A third plane ${\Pi }_3$ has the Cartesian equation $x+3y+kz=10$ where $k\in \mathrm{ℝ}$. The three planes do not meet at a unique point.

Find the exact value of $k$ and determine the geometrical relationship between the three planes.

Consider the four planes with Cartesian equations:

 ${\Pi }_1:6x-y+3z=16$

${\Pi }_2:4x+ky+2z=4$

${\Pi }_3:2x-5y+2z=7$

${\Pi }_4:x+3y-z=m$

where $k$ and $m$ are real constants.

In the case where there is no unique point of intersection of the three planes ${\Pi }_1,{\Pi }_2$and ${\Pi }_3$, find the value of $k$ and give a geometric interpretation of the three planes.

In the case where $k=6$, find the coordinates of the point of intersection between the three planes ${\Pi }_1,{\Pi }_2\mathrm{and}{\Pi }_3.$

In the case where there is a common line of intersection between the three planes ${\Pi }_2,{\Pi }_{3}and{\Pi }_4$, find the values of $k$ and $m$.

The point P(2, 0, -1)  is reflected in the plane $\Pi$ which has equation $\mathrm{r}·\left(\begin{array}{c}4\\ -3\\ 5\end{array}\right)=78.$

Find the coordinates of the reflection of P in the plane $\Pi$.

The line $L_1$ passes through the point P and intersects the plane $\Pi$ at the point Q(8, 3, 11) . The line $L_1$is reflected in the plane $\Pi$ to form line $L_2$.

Find a vector equation of the line $L_2$ .

Find the acute angle, in degrees, between the lines $L_1$and $L_2$.

Two planes are defined by the equations:

 ${\Pi }_1:x+2y-z=5,$

${\Pi }_2:2x+5y+2z=7.$

Find the exact value of $\cos \theta$ where $\theta$ is the acute angle between ${\Pi }_1$ and ${\Pi }_2$.

${\Pi }_1$ and ${\Pi }_2$ intersect at the line $L_{1,2}$. A third plane ${\Pi }_3$ is defined by the equation $x+ky+11z=m$ where $k,m\in \mathrm{ℝ}$ and ${\Pi }_3$ is perpendicular to ${\Pi }_1$.  When $m=a$ the line $L_{1,2}$lies on all three planes.

Find the values of $k$ and $a$.

Given that $m\ne a,{\Pi }_1$ and ${\Pi }_3$ intersect at the line $L_{1,3},{\Pi }_{2}and{\Pi }_3$ intersect at the line $L_{2,3}.$ The shortest distance between the lines $L_{1,2}\mathrm{and}{L}_{1,3}$ is $\sqrt{11}$.

Find the shortest distance between the lines $L_{1,2}$and $L_{2,3}$. Give your answer as an exact value.

The plane $\Pi$ is defined by the Cartesian equation $4x-5y+3z=59$. 

The line $L$ is defined by the Cartesian equation $\frac{4-x}{2}=y+1=2\left(z-3\right)$. 

Determine whether the point P(5, 8, 15) is closer to the plane $\Pi$ or the line $L$.

Fully justify your answer.