A line and a plane, two planes

Space geometry and vectors, grade 12 (4 units) Lesson 2 of 20Learning

A line and a plane, two planes

The positions of a line and a plane

A line and a plane are in one of three positions:

the line lies in the plane – all its points are in the plane. It is enough that two points of the line are in the plane.
the line meets the plane at one point.
the line is parallel to the plane – they have no common point.

In the box in the drawing, compared with the base plane $\displaystyle ABCD$ :

$\displaystyle BD$ lies in the plane: $\displaystyle B$ and $\displaystyle D$ are both in the base.
$\displaystyle AC'$ meets it at the point $\displaystyle A$ : the point $\displaystyle C'$ is above the base.
$\displaystyle A'C'$ is parallel to it: $\displaystyle A'C'$ is parallel to $\displaystyle AC$ , which lies in the base.

The last case is a handy way to recognise a line parallel to a plane: a line that is not in the plane and is parallel to some line in the plane is parallel to the plane.

Two planes

Two different planes are in one of two positions:

intersecting – they meet along a whole line, the line of intersection. The base plane $\displaystyle ABCD$ and the plane $\displaystyle ABB'A'$ meet along the line $\displaystyle AB$ .
parallel – they have no common point: the base $\displaystyle ABCD$ and the top $\displaystyle A'B'C'D'$ .

To find the line of intersection, look for two points that belong to both planes. The line through them is the line of intersection.

Example: positions in a box

In the box $\displaystyle ABCDA'B'C'D'$ :

$\displaystyle BB'$ is the line of intersection of the planes $\displaystyle ABB'A'$ and $\displaystyle BCC'B'$ : $\displaystyle B$ and $\displaystyle B'$ are in both.
The plane $\displaystyle ACC'A'$ meets the base plane along the line $\displaystyle AC$ .
$\displaystyle AD$ lies in the plane $\displaystyle ADD'A'$ .
$\displaystyle A'D'$ is parallel to the base plane: it is parallel to $\displaystyle AD$ .
$\displaystyle BD'$ meets the plane $\displaystyle A'B'C'D'$ at the point $\displaystyle D'$ .

Exercise 1

In the box $\displaystyle ABCDA'B'C'D'$ , what is the position of the line and the plane? Explain.

$\displaystyle C'D'$ and the base plane $\displaystyle ABCD$

$\displaystyle BD'$ and the plane $\displaystyle A'B'C'D'$

$\displaystyle B'C'$ and the plane $\displaystyle BCC'B'$

$\displaystyle A'C'$ and the plane $\displaystyle ACC'A'$

Exercise 2

In the same box, what is the position of the two planes? If they intersect, write the line of intersection.

$\displaystyle ABB'A'$ and $\displaystyle DCC'D'$

$\displaystyle ABCD$ and $\displaystyle BCC'B'$

$\displaystyle ABC'D'$ and $\displaystyle DCC'D'$

Exercise 3

In the same box:

Which faces of the box hold the point $\displaystyle A$ ?

Which faces of the box do not hold the point $\displaystyle A$ ?

Exercise 4

In the pyramid $\displaystyle SABCD$ , the base is the parallelogram $\displaystyle ABCD$ . What is the position? Explain.

the base plane and the plane $\displaystyle SAB$

the line $\displaystyle DC$ and the base plane

the line $\displaystyle DC$ and the plane $\displaystyle SAB$

the line $\displaystyle SC$ and the base plane

Formulas: space geometry and vectors, grade 12 (4 units)

Open as a page

The right-angled triangle and a triangle's area

Pythagoras’ theorem – $\displaystyle c$ the hypotenuse
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
$\displaystyle a$ the leg opposite the angle $\displaystyle \alpha$ , $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
$\displaystyle b$ the leg next to the angle $\displaystyle \alpha$
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
The leg opposite the angle over the leg next to it
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The cosine of an obtuse angle
$\displaystyle \cos(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
A triangle’s area – two sides and the angle between them
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

The prism

The surface area – the lateral surface and the two bases; $\displaystyle B$ the base’s area
$\displaystyle S$ $\displaystyle =$ $\displaystyle M+2B$
The lateral area of a right prism – $\displaystyle P$ the base’s perimeter, $\displaystyle h$ the height
$\displaystyle M$ $\displaystyle =$ $\displaystyle P\cdot h$
The volume of a prism – $\displaystyle h$ the height, the distance between the bases
$\displaystyle V$ $\displaystyle =$ $\displaystyle B\cdot h$

The pyramid

The surface area – the lateral surface and the base
$\displaystyle S$ $\displaystyle =$ $\displaystyle M+B$
The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{B\cdot h}{3}$

Vectors, geometrically

The triangle rule
$\displaystyle \overrightarrow{AB}+\overrightarrow{BC}$ $\displaystyle =$ $\displaystyle \overrightarrow{AC}$
The difference of two vectors leaving the same point
$\displaystyle \overrightarrow{AD}-\overrightarrow{AB}$ $\displaystyle =$ $\displaystyle \overrightarrow{BD}$
Collinear vectors
$\displaystyle \vec{v}=t\cdot\vec{u}$
A linear combination of $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$
$\displaystyle \vec{w}=s\vec{u}+t\vec{v}$

Vectors in coordinates

The vector between two points – the end minus the start
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)$
A vector’s length
$\displaystyle |\vec{u}|$ $\displaystyle =$ $\displaystyle \sqrt{x^2+y^2+z^2}$
The midpoint of the segment $\displaystyle AB$
$\displaystyle M=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2},\frac{z_A+z_B}{2}\right)$

The scalar product

The definition – $\displaystyle \alpha$ the angle between the vectors
$\displaystyle \vec{u}\cdot\vec{v}$ $\displaystyle =$ $\displaystyle |\vec{u}|\cdot|\vec{v}|\cdot\cos\alpha$
In coordinates
$\displaystyle \vec{u}\cdot\vec{v}$ $\displaystyle =$ $\displaystyle x_1x_2+y_1y_2+z_1z_2$
The angle between two vectors
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{\vec{u}\cdot\vec{v}}{|\vec{u}|\cdot|\vec{v}|}$
A vector’s length by the scalar product
$\displaystyle |\vec{u}|$ $\displaystyle =$ $\displaystyle \sqrt{\vec{u}^2}$
A short scalar product
$\displaystyle (\vec{u}+\vec{v})^2$ $\displaystyle =$ $\displaystyle \vec{u}^2+2\vec{u}\cdot\vec{v}+\vec{v}^2$
A short scalar product
$\displaystyle (\vec{u}+\vec{v})\cdot(\vec{u}-\vec{v})$ $\displaystyle =$ $\displaystyle \vec{u}^2-\vec{v}^2$
Perpendicular vectors
$\displaystyle \vec{u}\perp\vec{v}\iff\vec{u}\cdot\vec{v}=0$