Review: space geometry and vectors

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

Review: space geometry and vectors

1

$\displaystyle SABCD$ is a pyramid whose base is the parallelogram $\displaystyle ABCD$ .

a

Write two parallel lines that determine the base plane.

b

The line $\displaystyle AB$ is the line of intersection of which two planes?

c

What is the position of the line $\displaystyle DC$ and the plane $\displaystyle SAB$ ? And of the lines $\displaystyle DC$ and $\displaystyle SB$ ?

2

In the triangle $\displaystyle ABC$ : $\displaystyle {\overrightarrow{AB}}={\vec{u}}$ , $\displaystyle {\overrightarrow{AC}}={\vec{v}}$ .
$\displaystyle M$ is the midpoint of $\displaystyle AB$ , and $\displaystyle N$ the midpoint of $\displaystyle AC$ .

a

\displaystyle \overrightarrow{MN}

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b

\displaystyle \overrightarrow{BC}

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c

Prove with vectors: $\displaystyle MN\parallel BC$ and $\displaystyle {MN}={\frac{1}{2}BC}$ .

3

In the right prism $\displaystyle ABCDA'B'C'D'$ the base $\displaystyle ABCD$ is a parallelogram, and $\displaystyle E$ is the midpoint of $\displaystyle D'C'$ .
$\displaystyle {\overrightarrow{AB}}={\vec{u}}$ , $\displaystyle {\overrightarrow{BC}}={\vec{v}}$ , $\displaystyle {\overrightarrow{AA'}}={\vec{w}}$ .
$\displaystyle {|\vec{u}|}={|\vec{v}|}={8}$ , $\displaystyle {|\vec{w}|}={5}$ , $\displaystyle {|\overrightarrow{AE}|}={11}$ .
In lesson 15 you found $\displaystyle {\vec{u}\cdot\vec{v}}={16}$ .

a

The base’s area, exactly (hint:

\displaystyle {\sin\alpha}={\sqrt{1-\cos^2\alpha}}

)

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b

The prism’s volume, to two decimal places

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c

\displaystyle \overrightarrow{D'A}\cdot\overrightarrow{D'E}

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d

Is $\displaystyle {\angle AD'E}={90^{\circ}}$ ? Explain.

4

In the pyramid $\displaystyle SABCD$ the vertices are $\displaystyle A(1,1,0)$ , $\displaystyle B(5,1,0)$ , $\displaystyle C(5,4,0)$ , $\displaystyle D(1,4,0)$ , $\displaystyle S(3,2.5,6)$ .

a

Prove that the base is a rectangle.

b

The point $\displaystyle O(3,2.5,0)$ is where the diagonals meet. Prove that $\displaystyle SO$ is the pyramid’s height.

c

The pyramid’s volume

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d

The side edge

\displaystyle SA

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e

The angle

\displaystyle ASC

, to two decimal places

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5

$\displaystyle {\vec{u}}={(1,1,2)}$ , $\displaystyle {\vec{v}}={(2,-1,1)}$ , $\displaystyle {\vec{w}}={(5,-1,4)}$ .
Write $\displaystyle \vec{w}$ as a linear combination of $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ : write the system and solve it.

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

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