Practice: vectors in the geometric approach

Vectors, grade 12 (5 units) Lesson 11 of 23Practice

Practice: vectors in the geometric approach

$\displaystyle ABCD$ is a parallelogram, $\displaystyle {\overrightarrow{AB}}={\underline{u}}$ , $\displaystyle {\overrightarrow{AD}}={\underline{v}}$ .
$\displaystyle E$ is on $\displaystyle CD$ , and $\displaystyle {DE:EC}={1:2}$ . The segments $\displaystyle AE$ and $\displaystyle BD$ meet at the point $\displaystyle K$ .

\displaystyle \overrightarrow{AE}

/ – 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

Express $\displaystyle \overrightarrow{AK}$ in two ways, compare coefficients and solve.

In what ratio does $\displaystyle K$ divide $\displaystyle AE$ ?

In the triangle $\displaystyle ABC$ : $\displaystyle {\overrightarrow{AB}}={\underline{u}}$ , $\displaystyle {\overrightarrow{AC}}={\underline{v}}$ .
$\displaystyle D$ is the midpoint of $\displaystyle AB$ . $\displaystyle E$ is on $\displaystyle AC$ , and $\displaystyle {AE:EC}={2:1}$ . The segments $\displaystyle BE$ and $\displaystyle CD$ meet at the point $\displaystyle P$ .

\displaystyle {\overrightarrow{AP}}={\overrightarrow{AB}+s\cdot\overrightarrow{BE}}

/ – 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 {\overrightarrow{AP}}={\overrightarrow{AC}+t\cdot\overrightarrow{CD}}

/ – 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

Compare coefficients and solve.

Find the ratios $\displaystyle BP:PE$ and $\displaystyle CP:PD$ .

In the box $\displaystyle ABCDA'B'C'D'$ : $\displaystyle {\overrightarrow{AB}}={\underline{u}}$ , $\displaystyle {\overrightarrow{AD}}={\underline{v}}$ , $\displaystyle {\overrightarrow{AA'}}={\underline{w}}$ .
$\displaystyle K$ is the midpoint of $\displaystyle A'D'$ , and $\displaystyle L$ the centre of the face $\displaystyle BCC'B'$ .

\displaystyle \overrightarrow{AK}

/ – 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 \overrightarrow{AL}

/ – 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 \overrightarrow{KL}

/ – 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

Is $\displaystyle KL$ parallel to the plane $\displaystyle ABB'A'$ ? Explain.

In the cube $\displaystyle ABCDA'B'C'D'$ : $\displaystyle {\overrightarrow{AB}}={\underline{a}}$ , $\displaystyle {\overrightarrow{AD}}={\underline{b}}$ , $\displaystyle {\overrightarrow{AA'}}={\underline{c}}$ .
The face diagonals from $\displaystyle A$ : $\displaystyle {\underline{v_1}}={\overrightarrow{AC}}$ , $\displaystyle {\underline{v_2}}={\overrightarrow{AB'}}$ , $\displaystyle {\underline{v_3}}={\overrightarrow{AD'}}$ .

Is $\displaystyle \underline{v_3}$ linearly dependent on $\displaystyle \underline{v_1}$ and $\displaystyle \underline{v_2}$ ? Try $\displaystyle {\underline{v_3}}={s\underline{v_1}+t\underline{v_2}}$ , compare the coefficients of $\displaystyle \underline{a}$ , $\displaystyle \underline{b}$ , $\displaystyle \underline{c}$ , and solve.

Show that

\displaystyle {\frac{1}{2}(\underline{v_1}+\underline{v_2}+\underline{v_3})}={\overrightarrow{AC'}}

: write the left side with

\displaystyle \underline{a}

\displaystyle \underline{b}

\displaystyle \underline{c}

and simplify

/ – 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

$\displaystyle {\overrightarrow{OA}}={\underline{u}}$ , $\displaystyle {\overrightarrow{OB}}={\underline{v}}$ , $\displaystyle {\overrightarrow{OC}}={2\underline{u}+k\underline{v}}$ , where $\displaystyle \underline{u}$ and $\displaystyle \underline{v}$ are independent.
For which $\displaystyle k$ do the points $\displaystyle A$ , $\displaystyle B$ , $\displaystyle C$ lie on one line? Write $\displaystyle {\overrightarrow{AC}}={t\cdot\overrightarrow{AB}}$ , compare coefficients and solve.

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