Points and Vectors

2010015704

Level:

Given the vectors $ \vec{a} $, $ \vec{b} $, and $ \vec{c} $ shown in the picture, express the vector $ \vec{c} $ as a linear combination of vectors $ \vec{a} $ and $ \vec{b} $.

$ \vec{c} = -\vec{a}-2\vec{b} $

$ \vec{c} = -\vec{a} + \frac12 \vec{b} $

$ \vec{c} = -2\vec{a} - \vec{b} $

$ \vec{c} = 2\vec{a} + \frac32 \vec{b} $

The picture shows a rectangular cuboid $ ABCDEFGH $. In the cuboid find the vector that is the sum of $ \overrightarrow{AB} + \overrightarrow{AH} + \overrightarrow{EG} + \overrightarrow{FA} + \overrightarrow{HE} $.

$ \overrightarrow{AC} $

$ \overrightarrow{FH} $

$ \overrightarrow{AG} $

$ \overrightarrow{BH} $

2010015702

Level:

We are given the vectors $\vec{a}=(1;2;-3)$, $\vec{b}=(0;-1;2)$, and $\vec{c}=(-1;1;0)$. Find the mixed product $(\vec{a}\times\vec{b})\cdot\vec{c}$.

$ -3 $

The mixed product is not defined.

$ (0;-2;0) $

$ -2 $

2010015701

Level:

$ S $ is the midpoint of $ AB $. $ B = [3; -5] $, $ S = [0; -7] $. Find the coordinates of $ A $.

$ A = [-3; -9] $

$ A = [1.5; -6] $

$ A = [3; -12] $

$ A = [-3; -2] $

2010007410

Level:

Find the vector of length $\sqrt{2}$, which is perpendicular to the $y$-axis.

$ \left(\sqrt{2};0\right)$

$ (2;0)$

$ (0;2)$

$ \left(0;\sqrt{2}\right)$

2010007409

Level:

Find the vector of length $\sqrt{3}$, which is perpendicular to the $x$-axis.

$ \left(0;\sqrt{3}\right)$

$ (3;0)$

$ (0;3)$

$ \left(\sqrt{3};0\right)$

2010007408

Level:

Given the points $ A=[4;4]$ and $S=[-2;2]$, find the point $B$, such that $S$ is the center of $AB$.

$ B=[-8;0]$

$ B=[1;3]$

$ B=[10;6]$

$ B=[-5;1]$

2010007407

Level:

Given the points $ A=[3;1]$ and $S=[-1;3]$, find the point $B$, such that $S$ is the center of $AB$.

$ B=[-5;5]$

$ B=[1;2]$

$ B=[7;-1]$

$ B=[-3;4]$

2010007406

Level:

Find the parallel vector to the vector $\overrightarrow{AB}$, where $A=[-3;2]$, $B=[1;4]$.

$ (2;1)$

$ (-1;2)$

$ (4;6)$

$ (4;1)$

2010007405

Level:

Find the parallel vector to the vector $\overrightarrow{AB}$, where $A=[1;2]$, $B=[4;-1]$.

$ (1;-1)$

$ (3;3)$

$ (3;1)$

$ (5;1)$

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2010015704

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