Chapter 10 Basic Concepts of Descriptive Vector Geometry

Section 10.3 Final Test

10.3.1 Final Test Module 10

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Exercise 10.3.1
Specify the vectors that have the arrows shown in the figure below as their representatives. 1. Red vector: .
2. Purple vector: .
3. Blue vector: .
4. Green vector: .
5. Black vector: .
Vektoren können in der Form (a;b) eingegeben werden, zum Beispiel (8;-9) für den Vektor $\left(\begin{array}{c}\hfill 8\hfill \\ \hfill -9\hfill \end{array}\right)$.

Exercise 10.3.2

In still air conditions, a sports aircraft can fly with a velocity of 150 kilometres per hour due south. However, a crosswind blowing from the west with a velocity of 30 kilometres per hour causes the plane to drift. Represent the velocity of the aircraft as the sum of two vectors in the plane, where the second component corresponds to the north-south-direction (positive values for north) and the first component corresponds to the east-west-direction (positive values for east). Drop the unit of measure (kilometres per hour) in your calculation:
1. In still air conditions, the velocity is .
2. The wind causes an additional velocity of .
3. The drifting aircraft has in total the velocity vector .
4. The length of this vector (absolute value of the velocity) is .
Vektoren können in der Form (a;b) eingegeben werden, zum Beispiel (8;-9) für den Vektor $\left(\begin{array}{c}\hfill 8\hfill \\ \hfill -9\hfill \end{array}\right)$.

Exercise 10.3.3
Let three points $P=\left(3;4\right)$, $Q=\left(1;0\right)$, and $R=\left(-2;1\right)$ be given in the plane. Calculate the following vectors:
1. $\stackrel{\to }{PQ}$$=$ .
2. $\stackrel{\to }{QR}$$=$ .
3. $\stackrel{\to }{RR}$$=$ .
4. $\stackrel{\to }{QP}$$=$ .
5. $\stackrel{\to }{RP}$$=$ .

Exercise 10.3.4
Let three points $P=\left(1;2;3\right)$, $Q=\left(3;0;0\right)$, and $R=\left(-1;2;2\right)$ be given in space. Calculate the following vectors:
1. $\stackrel{\to }{PQ}$$=$ .
2. $\stackrel{\to }{RQ}$$=$ .

Find the position vector $\stackrel{\to }{M}$ of the midpoint $M$ of the line segment $\stackrel{‾}{PR}$$\stackrel{\to }{M}$$=$
.

Exercise 10.3.5

Find the intersection point $S$ of the two lines given by the equations in vector form

$\stackrel{\to }{r}\mathrm{ }=\mathrm{ }\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 2\hfill \end{array}\right)+\alpha ·\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right) ; \alpha \in ℝ \text{and} \stackrel{\to }{r}\mathrm{ }=\mathrm{ }\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill -2\hfill \end{array}\right)+\beta ·\left(\begin{array}{c}\hfill -2\hfill \\ \hfill -4\hfill \\ \hfill 4\hfill \end{array}\right) ; \beta \in ℝ .$

1. The position vector of the intersection point is $\stackrel{\to }{S}=$ .
2. It results from the first equation, for the parameter value $\alpha$$=$ .
3. It results from the second equation, for the parameter value $\beta$$=$ .

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