The axes in space

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

The axes in space

Three axes

Space has three axes, perpendicular to each other, meeting at the origin $\displaystyle O$ : the $\displaystyle x$ -axis, the $\displaystyle y$ -axis and the $\displaystyle z$ -axis. In the drawing the $\displaystyle x$ -axis comes out of the page towards you, the $\displaystyle y$ -axis goes right, and the $\displaystyle z$ -axis up.

The coordinates of the point $\displaystyle P(x,y,z)$ say how to get there from $\displaystyle O$ : $\displaystyle x$ units along the $\displaystyle x$ -axis, then $\displaystyle y$ units along the $\displaystyle y$ -axis, and last $\displaystyle z$ units up. In the drawing: $\displaystyle P(3,4,2)$ .

A negative number means moving against the axis.

The coordinate planes and the axes

Every two axes determine a plane:

the plane $\displaystyle xy$ – its points are $\displaystyle (x,y,0)$ : $\displaystyle {z}={0}$ .
the plane $\displaystyle xz$ – $\displaystyle (x,0,z)$ : $\displaystyle {y}={0}$ .
the plane $\displaystyle yz$ – $\displaystyle (0,y,z)$ : $\displaystyle {x}={0}$ .

And on the axes themselves two of the coordinates are 0:

on the $\displaystyle x$ -axis – $\displaystyle (x,0,0)$ ; on the $\displaystyle y$ -axis – $\displaystyle (0,y,0)$ ; on the $\displaystyle z$ -axis – $\displaystyle (0,0,z)$ .

The point below $\displaystyle P(3,4,2)$ in the plane $\displaystyle xy$ is $\displaystyle (3,4,0)$ , and $\displaystyle P$ is 2 away from the plane $\displaystyle xy$ .

The distance from the origin

The dashed lines in the drawing are edges of a box whose dimensions are $\displaystyle x$ , $\displaystyle y$ , $\displaystyle z$ , and $\displaystyle OP$ is its diagonal. So, by Pythagoras’ theorem twice:

\displaystyle OP

\displaystyle =

\displaystyle \sqrt{x^2+y^2+z^2}

For example, for $\displaystyle P(3,4,2)$ : $\displaystyle {OP}={\sqrt{9+16+4}}={\sqrt{29}}$ .

Exercise 1

The box $\displaystyle OABCO'A'B'C'$ has a vertex at the origin, and three edges on the axes: $\displaystyle {OA}={4}$ , $\displaystyle {OC}={5}$ , $\displaystyle {OO'}={3}$ .
Write the coordinates of the eight vertices.

Exercise 2

For each point: is it on an axis, in the plane of two axes, or in neither? Say which.

$\displaystyle (0,3,0)$

$\displaystyle (2,0,5)$

$\displaystyle (0,0,-4)$

$\displaystyle (1,2,3)$

Exercise 3

The point $\displaystyle Q(3,-2,4)$ is given.

Which point of the plane $\displaystyle xy$ is right below (or above) $\displaystyle Q$ ?

The distance of

\displaystyle Q

from the plane

\displaystyle xy

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The distance of

\displaystyle Q

from the origin, exactly

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Find the distance of the point $\displaystyle P(2,3,6)$ from the origin.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

The base of the right pyramid $\displaystyle OACBS$ is the rectangle $\displaystyle OACB$ , in the plane $\displaystyle xy$ : $\displaystyle A(4,0,0)$ , $\displaystyle B(0,8,0)$ .
The height $\displaystyle SD$ is perpendicular to the base plane, and $\displaystyle {SD}={12}$ .

What are the coordinates of $\displaystyle C$ ?

Why is $\displaystyle D$ where the base’s diagonals meet? What are its coordinates?

What are the coordinates of $\displaystyle S$ ?

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
$\displaystyle \vec{u}\perp\vec{v}\iff\vec{u}\cdot\vec{v}=0$