Dot Product of Vectors
Points and vectors
2010015707
Level:
A
Given vectors \(\vec{a} = (-1;2;1)\),
\(\vec{b} = (0;-1;1)\), and
\(\vec{c} = (-2;0;1)\), find the length
of the vector \(\vec{u} =\vec{ a} - 2\vec{b} + \vec{c}\).
\(|\vec{u}| = 5\)
\(|\vec{u}| = \sqrt{10}\)
\(|\vec{u}| = 3\)
\(|\vec{u}| = 1\)
2010015706
Level:
A
Given the vectors \(\vec{a}\),
\(\vec{b}\), and
\(\vec{c}\), find
\(\vec{a} - 3\vec{b} +\vec{ c}\).
\((10;5)\)
\((-8;5)\)
\((-8;-1)\)
\((7;5)\)
2010015705
Level:
A
We are given points \( A = [2;1] \), \( B = [7;2] \), and \( T = [4;3] \), where point \( T \) is the centroid of triangle \( ABC \). Find the coordinates of \( C \), which is the vertex of \( ABC \).
\( C = [3;6] \)
\( C = [4;8] \)
\( C = [3.5;7] \)
\( C = [5;6] \)
2010015704
Level:
A
Given the vectors \( \overrightarrow{a} \), \( \overrightarrow{b} \), and \( \overrightarrow{c} \) shown in the picture, express the vector \( \overrightarrow{c} \) as a linear combination of vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \).
\( \overrightarrow{c} = -\overrightarrow{a}-2\overrightarrow{b} \)
\( \overrightarrow{c} = -\overrightarrow{a} + \frac12 \overrightarrow{b} \)
\( \overrightarrow{c} = -2\overrightarrow{a} - \overrightarrow{b} \)
\( \overrightarrow{c} = 2\overrightarrow{a} + \frac32 \overrightarrow{b} \)
2010015703
Level:
A
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:
C
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:
A
\( S \) 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:
A
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:
A
Find the vector of length \(\sqrt{3}\), which is perpendicular to the \(x\)-axis.
\( \left(0;\sqrt{3}\right)\)
\( (3;0)\)
\( (0;3)\)
\( \left(\sqrt{3};0\right)\)