Vectors in axes

Vectors, grade 12 (5 units) Lesson 15 of 23Learning

Vectors in axes

Axes in space

Three mutually perpendicular axes meet at the origin $\displaystyle O$ : the $\displaystyle x$ -axis (towards us), the $\displaystyle y$ -axis (to the right) and the $\displaystyle z$ -axis (up).

Every point has three coordinates: $\displaystyle P(3,4,3)$ – go 3 along $\displaystyle x$ , 4 along $\displaystyle y$ and 3 up.
The coordinate planes: the $\displaystyle xy$ plane (where $\displaystyle {z}={0}$ ), the $\displaystyle xz$ plane ( $\displaystyle {y}={0}$ ), the $\displaystyle yz$ plane ( $\displaystyle {x}={0}$ ).
The position vector of $\displaystyle P$ is $\displaystyle {\overrightarrow{OP}}={(3,4,3)}$ . If $\displaystyle \underline{i}$ , $\displaystyle \underline{j}$ , $\displaystyle \underline{k}$ are vectors of length 1 along the axes, $\displaystyle {\overrightarrow{OP}}={3\underline{i}+4\underline{j}+3\underline{k}}$ – the unique representation of $\displaystyle \overrightarrow{OP}$ by them.

A vector whose tail is at the origin describes a point, so the same coordinates describe both a vector and a point.

Operations in coordinates

Since $\displaystyle {\underline{u}}={u_1\underline{i}+u_2\underline{j}+u_3\underline{k}}$ , the operations are done coordinate by coordinate:

$\displaystyle {(u_1,u_2,u_3)+(v_1,v_2,v_3)}={(u_1+v_1,\ u_2+v_2,\ u_3+v_3)}$ , and $\displaystyle {t(u_1,u_2,u_3)}={(tu_1,tu_2,tu_3)}$ ;
a vector between two points – the end minus the start: $\displaystyle {\overrightarrow{AB}}={B-A}$ ;
length: $\displaystyle {|\underline{u}|}={\sqrt{u_1^2+u_2^2+u_3^2}}$ (Pythagoras twice, as in the box’s diagonal);
a point $\displaystyle M$ dividing $\displaystyle AB$ in the ratio $\displaystyle m:n$ : $\displaystyle {M}={\frac{nA+mB}{m+n}}$ ; the midpoint – $\displaystyle \frac{A+B}{2}$ .

The scalar product in coordinates

The unit vectors are perpendicular and of length 1: $\displaystyle {\underline{i}^2}={\underline{j}^2}={\underline{k}^2}={1}$ , and $\displaystyle {\underline{i}\cdot\underline{j}}={\underline{i}\cdot\underline{k}}={\underline{j}\cdot\underline{k}}={0}$ . Opening the brackets leaves only the products of matching coordinates:

\displaystyle \underline{u}\cdot\underline{v}

\displaystyle =

\displaystyle u_1v_1+u_2v_2+u_3v_3

Example: a triangle in coordinates

The points $\displaystyle A(1,2,3)$ , $\displaystyle B(4,0,-1)$ , $\displaystyle C(2,5,1)$ are given.
Find the angle $\displaystyle BAC$ and the midpoint of the side $\displaystyle BC$ .

$\displaystyle {\overrightarrow{AB}}={(3,-2,-4)}$ , $\displaystyle {\overrightarrow{AC}}={(1,3,-2)}$ .
$\displaystyle {\overrightarrow{AB}\cdot\overrightarrow{AC}}={3-6+8}={5}$ .
$\displaystyle {|\overrightarrow{AB}|}={\sqrt{9+4+16}}={\sqrt{29}}$ , $\displaystyle {|\overrightarrow{AC}|}={\sqrt{1+9+4}}={\sqrt{14}}$ .
$\displaystyle {\cos\angle BAC}={\frac{5}{\sqrt{29}\cdot\sqrt{14}}}$ , so $\displaystyle \angle BAC\approx75.63^{\circ}$ .
The midpoint of $\displaystyle BC$ : $\displaystyle {\left(\frac{4+2}{2},\frac{0+5}{2},\frac{-1+1}{2}\right)}={\left(3,\frac{5}{2},0\right)}$ .

Exercise 1

$\displaystyle {\underline{u}}={(2,-1,3)}$ , $\displaystyle {\underline{v}}={(1,4,-2)}$ .

\displaystyle 2\underline{u}-3\underline{v}

/ – 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

\displaystyle \underline{u}\cdot\underline{v}

/ – 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

\displaystyle |\underline{u}|

/ – 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

\displaystyle \underline{u}

and

\displaystyle \underline{v}

, rounded to two 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 2

The points $\displaystyle A(1,0,2)$ and $\displaystyle B(4,6,-1)$ are given.
Find the point $\displaystyle M$ on $\displaystyle AB$ for which $\displaystyle {AM:MB}={1:2}$ .

/ – 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

$\displaystyle {\underline{a}}={(1,2,3)}$ , $\displaystyle {\underline{b}}={(4,5,6)}$ , $\displaystyle {\underline{c}}={(2,1,0)}$ .

Express $\displaystyle \underline{c}$ as a linear combination of $\displaystyle \underline{a}$ and $\displaystyle \underline{b}$ : solve the vector equation, which is three equations.

\displaystyle \frac{1}{3}\underline{a}+\frac{1}{3}\underline{c}

/ – 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

\displaystyle \frac{1}{3}\underline{b}-\frac{1}{3}\underline{a}

/ – 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

You found two different representations of $\displaystyle (1,1,1)$ with $\displaystyle \underline{a}$ , $\displaystyle \underline{b}$ , $\displaystyle \underline{c}$ . Does this contradict the uniqueness of a representation? Explain.

Exercise 4

Find $\displaystyle a$ for which the vectors $\displaystyle (a,2,-1)$ and $\displaystyle (3,a,4)$ are perpendicular.

Exercise 5

The points $\displaystyle A(2,1,0)$ , $\displaystyle B(4,3,1)$ , $\displaystyle C(1,3,-2)$ are given.

\displaystyle \overrightarrow{AB}\cdot\overrightarrow{AC}

/ – 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 area of the triangle

\displaystyle ABC

/ – 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

Formulas: vectors, grade 12 (5 units)

Open as a page

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
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{|\underline{n_1}\cdot\underline{n_2}|}{|\underline{n_1}|\cdot|\underline{n_2}|}$