Practice: the scalar product, perpendicularity and volumes

Space geometry and vectors, grade 12 (4 units) Lesson 19 of 20Practice

Practice: the scalar product, perpendicularity and volumes

$\displaystyle {|\vec{u}|}={6}$ , $\displaystyle {|\vec{v}|}={4}$ , and the angle between them is $\displaystyle 120^{\circ}$ .
Find exactly:

\displaystyle \vec{u}\cdot\vec{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 |\vec{u}+\vec{v}|

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\displaystyle (\vec{u}+\vec{v})\cdot(\vec{u}-\vec{v})

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In the pyramid $\displaystyle ABCD$ the vertices are $\displaystyle A(-1,1,0)$ , $\displaystyle B(0,1,1)$ , $\displaystyle C(-4,3,-1)$ , $\displaystyle D(0,6,0)$ .
$\displaystyle DO$ is the pyramid’s height.

Prove that the pyramid is right: find $\displaystyle DA$ , $\displaystyle DB$ , $\displaystyle DC$ .

The angle

\displaystyle BAC

, to two decimal places

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The radius

\displaystyle R

of the circle through

\displaystyle A

\displaystyle B

\displaystyle C

(the law of sines), to two decimal places

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Why is $\displaystyle DO\perp AO$ ? And why is $\displaystyle {AO}={R}$ ?

The height

\displaystyle DO

, to two decimal places

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In the triangle $\displaystyle ABC$ the medians meet at the point $\displaystyle M$ .
$\displaystyle A(3,-4,1)$ , $\displaystyle B(4,1,8)$ , $\displaystyle C(2,0,7)$ .

Find the coordinates of $\displaystyle M$ .

How many times the area of the triangle $\displaystyle BMC$ is the area of $\displaystyle ABC$ ? Explain.

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The base of the right pyramid $\displaystyle OACBS$ is the rectangle $\displaystyle OACB$ : $\displaystyle A(4,0,0)$ , $\displaystyle B(0,8,0)$ , and $\displaystyle S(2,4,12)$ .
In a right pyramid each lateral face is an isosceles triangle, so its height goes down to the midpoint of the base edge.

The area of the face

\displaystyle SOA

, exactly (hint: the midpoint of

\displaystyle OA

\displaystyle (2,0,0)

)

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The area of the face

\displaystyle SAC

, exactly

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The lateral area, to two decimal places

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A cube of edge 3 sits on the axes, one of its vertices at the origin.
$\displaystyle O(0,0,0)$ , $\displaystyle P(3,3,3)$ , $\displaystyle Q(3,0,0)$ , $\displaystyle R(0,3,3)$ : $\displaystyle OP$ and $\displaystyle QR$ are two of the cube’s diagonals.
What is the acute angle between two diagonals of a cube? Round to two decimal places.

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In the prism $\displaystyle ABCDA'B'C'D'$ the base $\displaystyle ABCD$ is a parallelogram: $\displaystyle A(1,2,0)$ , $\displaystyle B(3,2,0)$ , $\displaystyle D(1,5,0)$ , $\displaystyle A'(2,3,4)$ .
The base is in the plane $\displaystyle xy$ .

Is the prism right? Explain with a scalar product.

What is the prism’s height? (Hint: the distance of $\displaystyle A'$ from the plane $\displaystyle xy$ .)

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The prism’s volume

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