Angles in space

Vectors, grade 12 (5 units) Lesson 2 of 23Learning

Angles in space

Projection and slanted line

The projection of a point $\displaystyle P$ on a plane is the foot $\displaystyle P'$ of the perpendicular from $\displaystyle P$ to the plane.
A line that cuts the plane and is not perpendicular to it is called slanted.
The projection of the slanted line $\displaystyle PA$ is the segment $\displaystyle P'A$ : it joins the meeting point $\displaystyle A$ to the projection of $\displaystyle P$ .
The distance of $\displaystyle P$ from the plane is $\displaystyle PP'$ – the shortest segment from $\displaystyle P$ to the plane.

The angle between a line and a plane

The angle between a line and a plane is the angle between the line and its projection on the plane – the angle $\displaystyle \alpha$ in the drawing.

It is between $\displaystyle 0^{\circ}$ and $\displaystyle 90^{\circ}$ ;
a line perpendicular to the plane makes an angle of $\displaystyle 90^{\circ}$ with it, and a parallel line, or one lying in it, an angle of $\displaystyle 0^{\circ}$ .

The triangle $\displaystyle PP'A$ is right-angled at $\displaystyle P'$ , since $\displaystyle PP'$ is perpendicular to the plane. So $\displaystyle \alpha$ is found in it: $\displaystyle {\tan\alpha}={\frac{PP'}{P'A}}$ .

The three perpendiculars theorem

A line in the plane is perpendicular to a slanted line if and only if it is perpendicular to the projection of the slanted line on the plane.

The three perpendiculars are the perpendicular $\displaystyle PP'$ to the plane, the perpendicular from $\displaystyle P'$ to the line in the plane, and the perpendicular from $\displaystyle P$ to the same line – and they meet at the same point. We prove the theorem with vectors in the lesson on using the scalar product.

The angle between two planes

Two planes meet along their line of intersection. From a point on that line, draw a perpendicular to it in each of the planes. The angle between the planes is the angle between these two perpendiculars.

In a right pyramid on a square base: the angle between the face $\displaystyle SBC$ and the base is $\displaystyle {\alpha}={\angle SMO}$ , where $\displaystyle M$ is the midpoint of $\displaystyle BC$ .

$\displaystyle OM\perp BC$ , since $\displaystyle OM$ is parallel to $\displaystyle AB$ ;
$\displaystyle SM\perp BC$ , by the three perpendiculars theorem: $\displaystyle SO$ is perpendicular to the base, and $\displaystyle OM$ is the projection of $\displaystyle SM$ .

The angle between skew lines

The angle between two skew lines is the angle between one of them and a line parallel to the other that meets it.

Example. In a cube, what is the angle between $\displaystyle AB'$ and $\displaystyle BC'$ ?

$\displaystyle BC'\parallel AD'$ (the quadrilateral $\displaystyle ABC'D'$ is a rectangle), so the angle is $\displaystyle \angle B'AD'$ .
$\displaystyle AB'$ , $\displaystyle AD'$ and $\displaystyle B'D'$ are diagonals of the cube’s faces, so they are equal: the triangle $\displaystyle AB'D'$ is equilateral.
The angle between the lines is $\displaystyle 60^{\circ}$ .

Example: the box’s diagonal and its base

In the box $\displaystyle ABCDA'B'C'D'$ : $\displaystyle {AB}={3}$ , $\displaystyle {AD}={4}$ , $\displaystyle {AA'}={6}$ .
Find the angle between the diagonal $\displaystyle AC'$ and the base’s plane.

The solution below finds the projection of $\displaystyle AC'$ on the base, and works out the angle in a right-angled triangle.

$\displaystyle CC'$ is perpendicular to the base, so the projection of $\displaystyle C'$ is $\displaystyle C$ , and the projection of $\displaystyle AC'$ is $\displaystyle AC$ : the angle is $\displaystyle {\alpha}={\angle C'AC}$ .
$\displaystyle {AC}={\sqrt{3^2+4^2}}={5}$ .
In the triangle $\displaystyle ACC'$ , right-angled at $\displaystyle C$ : $\displaystyle {\tan\alpha}={\frac{6}{5}}$ , so $\displaystyle \alpha\approx50.19^{\circ}$ .

Example: a face and the base

In a right pyramid $\displaystyle SABCD$ on a square base: the base’s side is 6, and the height $\displaystyle {SO}={4}$ .
Find the angle between the face $\displaystyle SBC$ and the base.

The solution below uses the angle $\displaystyle \angle SMO$ found above, where $\displaystyle M$ is the midpoint of $\displaystyle BC$ .

$\displaystyle OM$ is half the base’s side: $\displaystyle {OM}={3}$ .
In the triangle $\displaystyle SOM$ , right-angled at $\displaystyle O$ : $\displaystyle {\tan\alpha}={\frac{SO}{OM}}={\frac{4}{3}}$ , so $\displaystyle \alpha\approx53.13^{\circ}$ .

Exercise 1

The edge of a cube $\displaystyle ABCDA'B'C'D'$ is 2 long.
Find the angle between the diagonal $\displaystyle AC'$ and the base’s plane, rounded to two decimal places.

/ – 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

Exercise 2

In a right pyramid $\displaystyle SABCD$ on a square base: the base’s side is 6, and the height $\displaystyle {SO}={4}$ .
Find the angle between the lateral edge $\displaystyle SA$ and the base’s plane, rounded to two decimal places.

/ – 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 3

In the box $\displaystyle ABCDA'B'C'D'$ : $\displaystyle {AB}={4}$ , $\displaystyle {AD}={3}$ , $\displaystyle {AA'}={5}$ .

The angle between the plane

\displaystyle ABC'D'

and the base’s plane

/ – 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 angle between the skew lines

\displaystyle AA'

and

\displaystyle BD'

/ – 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

Explain: why is the angle you found in part 1 the angle $\displaystyle \angle C'BC$ ?

Formulas: vectors, grade 12 (5 units)

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Solids in space

The volume of a prism and a cylinder – $\displaystyle S$ the base’s area, $\displaystyle h$ the solid’s height
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\cdot h$
The volume of a pyramid and a cone – $\displaystyle S$ the base’s area, $\displaystyle h$ the solid’s height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\cdot h}{3}$
The lateral area of a right cone – $\displaystyle R$ the base’s radius, $\displaystyle \ell$ the slant height
$\displaystyle M$ $\displaystyle =$ $\displaystyle \pi R\ell$
The volume of a sphere – $\displaystyle R$ its radius
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4}{3}\pi R^3$
The surface area of a sphere
$\displaystyle M$ $\displaystyle =$ $\displaystyle 4\pi R^2$
The diagonal of a box of dimensions $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2+c^2}$

Trigonometry in the triangle

The law of sines – $\displaystyle R$ the radius of the circumscribed circle
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$
The area of a triangle – two sides and the angle between them
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

Vectors in the geometric approach

The triangle rule
$\displaystyle \overrightarrow{AB}+\overrightarrow{BC}$ $\displaystyle =$ $\displaystyle \overrightarrow{AC}$
A vector between two points – the end minus the start
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle \overrightarrow{OB}-\overrightarrow{OA}$
Collinear (parallel) vectors
$\displaystyle \underline{v}=t\cdot\underline{u}$
The point $\displaystyle M$ divides the segment $\displaystyle AB$ in the ratio $\displaystyle m:n$
$\displaystyle \overrightarrow{OM}=\frac{n}{m+n}\overrightarrow{OA}+\frac{m}{m+n}\overrightarrow{OB}$ $\displaystyle AM:MB=m:n$
$\displaystyle M$ the midpoint of the segment $\displaystyle AB$
$\displaystyle \overrightarrow{OM}=\frac{1}{2}\left(\overrightarrow{OA}+\overrightarrow{OB}\right)$

The scalar product

The length of a vector $\displaystyle {\underline{u}}={(u_1,u_2,u_3)}$
$\displaystyle |\underline{u}|=\sqrt{\underline{u}\cdot\underline{u}}=\sqrt{u_1^2+u_2^2+u_3^2}$
The scalar product in coordinates
$\displaystyle \underline{u}\cdot\underline{v}$ $\displaystyle =$ $\displaystyle u_1v_1+u_2v_2+u_3v_3$
The scalar product – $\displaystyle \alpha$ the angle between the vectors
$\displaystyle \underline{u}\cdot\underline{v}$ $\displaystyle =$ $\displaystyle |\underline{u}|\cdot|\underline{v}|\cdot\cos\alpha$
Perpendicular vectors
$\displaystyle \underline{u}\perp\underline{v}\iff\underline{u}\cdot\underline{v}=0$
A short scalar product
$\displaystyle (\underline{u}+\underline{v})^2$ $\displaystyle =$ $\displaystyle \underline{u}^2+2\underline{u}\cdot\underline{v}+\underline{v}^2$

Points and vectors in axes

A vector between two points
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)$
The distance between two points
$\displaystyle AB$ $\displaystyle =$ $\displaystyle \sqrt{(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}$
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 line and the plane

A line – $\displaystyle \underline{v}$ a point on the line, $\displaystyle \underline{u}$ the direction vector
$\displaystyle \underline{x}=\underline{v}+t\cdot\underline{u}$
A plane in parametric form – $\displaystyle \underline{u}$ , $\displaystyle \underline{v}$ not parallel
$\displaystyle \underline{x}=\underline{p}+t\cdot\underline{u}+s\cdot\underline{v}$
A plane’s equation – $\displaystyle {\underline{n}}={(a,b,c)}$ the normal vector
$\displaystyle ax+by+cz+d=0$
The plane through $\displaystyle P(x_1,y_1,z_1)$ whose normal vector is $\displaystyle (a,b,c)$
$\displaystyle a(x-x_1)+b(y-y_1)+c(z-z_1)=0$

Distances

The distance of a point $\displaystyle P(x_1,y_1,z_1)$ from the plane $\displaystyle {ax+by+cz+d}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|a\cdot x_1+b\cdot y_1+c\cdot z_1+d|}{\sqrt{a^2+b^2+c^2}}$
The distance between the parallel planes $\displaystyle {ax+by+cz+d_1}={0}$ and $\displaystyle {ax+by+cz+d_2}={0}$
$\displaystyle d$ $\displaystyle =$ $\displaystyle \frac{|d_1-d_2|}{\sqrt{a^2+b^2+c^2}}$

Angles

The angle between two lines – $\displaystyle \underline{u}$ , $\displaystyle \underline{v}$ the direction vectors
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{|\underline{u}\cdot\underline{v}|}{|\underline{u}|\cdot|\underline{v}|}$
The angle between the line $\displaystyle {\underline{x}}={\underline{v}+t\cdot\underline{u}}$ and the plane whose normal is $\displaystyle \underline{n}$
$\displaystyle \sin\beta$ $\displaystyle =$ $\displaystyle \frac{|\underline{n}\cdot\underline{u}|}{|\underline{n}|\cdot|\underline{u}|}$
The angle between two planes – $\displaystyle \underline{n_1}$ , $\displaystyle \underline{n_2}$ the normals
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{|\underline{n_1}\cdot\underline{n_2}|}{|\underline{n_1}|\cdot|\underline{n_2}|}$