Points and Vectors

1003030604

Level: 
B
Let \( \vec{a}=(2,- 3) \) and \( \vec{b}=(3,-2) \). Find all the vectors \( \vec{c} \) such that \[ \vec{a}\cdot\vec{c}=8\ \text{ and }\ \vec{b}\cdot\vec{c}=27. \]
\( \vec{c}=(13,6) \)
\( \vec{c_1}=(13,6),\ \vec{c_2}=(-13,-6) \)
\( \vec{c}=(13k,6k),\ k\in\mathbb{R} \)
\( \vec{c}=(-13,-6) \)

1003030605

Level: 
B
Let \( \vec{a}=(3,-5) \) and \( \vec{b}=(6,-10) \). Find all the vectors \( \vec{c} \) such that \[ \vec{a}\cdot\vec{c}=11\ \text{ and }\ \vec{b}\cdot\vec{c}=22\text{ .} \]
\( \vec{c}=(2+5k,-1+3k),\ k\in\mathbb{R} \)
\( \vec{c}_1=(7,2),\ \vec{c}_2=(-7,-2) \)
\( \vec{c}=(2k,-k),\ k\in\mathbb{R} \)
\( \vec{c}_1=(2,-1),\ \vec{c}_2=(-2,1) \)

1103021001

Level: 
B
Let \( ABCDEF \) be a regular hexagon with the centre \( S \) and the side of length \( 3\,\mathrm{cm}\). The point \( G \) is the midpoint of the segment \( AB \). The vectors \( \vec{u} \), \( \vec{v} \), \( \vec{w} \), \( \vec{z} \) are indicated in the hexagon shown in the picture. Find the dot product of: \( \vec{v}\cdot\vec{w} \), \( \vec{v}\cdot\vec{z} \) and \( \vec{v}\cdot\vec{u} \).
\( \vec{v}\cdot\vec{w}=9 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=27 \)
\( \vec{v}\cdot\vec{w}=9 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=9\sqrt6 \)
\( \vec{v}\cdot\vec{w}=\frac92 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=9\sqrt6 \)
\( \vec{v}\cdot\vec{w}=\frac92 \), \( \vec{v}\cdot\vec{z} = 1 \), \( \vec{v}\cdot\vec{u}=27 \)

1103030501

Level: 
B
The vectors \( \vec{u} \), \( \vec{v}\), \( \vec{w} \), \( \vec{z} \) are indicated in a cube shown in the figure. The cube edge length is \( 1 \). Find the dot products of: \[ \vec{v}\cdot\vec{z}\text{ ,}\ \ \vec{u}\cdot\vec{v} \text{ ,}\ \ \vec{w}\cdot\vec{u}\]
\( \vec{v}\cdot\vec{z}=1 \), \( \vec{u}\cdot\vec{v}=0 \), \( \vec{w}\cdot\vec{u}=1 \)
\( \vec{v}\cdot\vec{z}=\frac{\sqrt2}2 \), \( \vec{u}\cdot\vec{v}=1 \), \( \vec{w}\cdot\vec{u}=\sqrt3 \)
\( \vec{v}\cdot\vec{z}=\sqrt2 \), \( \vec{u}\cdot\vec{v}=0 \), \( \vec{w}\cdot\vec{u}=1 \)
\( \vec{v}\cdot\vec{z}=1 \), \( \vec{u}\cdot\vec{v}=1 \), \( \vec{w}\cdot\vec{u}=\sqrt3 \)

1103030502

Level: 
B
Find the coordinates of the vectors \( \vec{u} \) and \( \vec{v} \) given by the picture and evaluate their dot product.
\( \vec{u}=(-3,6),\ \ \vec{v} =(-9,-6),\ \ \vec{u}\cdot\vec{v} = -9 \)
\( \vec{u}=(3,-6),\ \ \vec{v} =(9,6),\ \ \vec{u}\cdot\vec{v} = -9 \)
\( \vec{u}=(-3,6),\ \ \vec{v} =(-9,-6),\ \ \vec{u}\cdot\vec{v} = 9 \)
\( \vec{u}=(3,-6),\ \ \vec{v} =(9,6),\ \ \vec{u}\cdot\vec{v} = 0 \)

1103030503

Level: 
B
Find the coordinates of the vectors \( \vec{u} \) and \( \vec{v} \) given by the picture and evaluate their dot product.
\( \vec{u}=(-8,-7,9),\ \ \vec{v} =(8,7,9),\ \ \vec{u}\cdot\vec{v} = -32 \)
\( \vec{u}=(-8,-7,9),\ \ \vec{v} =(8,7,9),\ \ \vec{u}\cdot\vec{v} = 0 \)
\( \vec{u}=(-8,-7,9),\ \ \vec{v} =(8,7,9),\ \ \vec{u}\cdot\vec{v} = (-64,-49,81) \)
\( \vec{u}=(8,7,-9),\ \ \vec{v} =(-8,-7,-9),\ \ \vec{u}\cdot\vec{v} = (-64,-49,81) \)

1103030504

Level: 
B
The vectors \( \vec{u} \) and \( \vec{v} \) are given by the figure. Find cosine of the angle \(\varphi \) between \( \vec{u} \) and \( \vec{v} \). Help: Use the dot product of the given vectors.
\( \cos\varphi=\frac{13\sqrt{10}}{50} \)
\( \cos\varphi=\frac{970}{50} \)
\( \cos\varphi=\frac{3\sqrt{10}}{10} \)
\( \cos\varphi=\frac{\sqrt{10}}{5} \)

1103030505

Level: 
B
The vectors \( \vec{u} \) and \( \vec{v} \) are given by the figure. Find cosine of the angle \( \varphi \) between \( \vec{u} \) and \( \vec{v} \). Help: Use the dot product of the given vectors.
\( \cos\varphi=-\frac9{17} \)
\( \cos\varphi=\frac9{17} \)
\( \cos\varphi=\frac{\sqrt{17}}{2\sqrt{13}} \)
\( \cos\varphi=-\frac{\sqrt{17}}{2\sqrt{13}} \)

1103030601

Level: 
B
In the cube \( ABCDEFGH \) find the angle \( \varphi \) between the vectors \( \vec{b}=\overrightarrow{EB} \) and \( \vec{a}=\overrightarrow{AK} \), where \( K \) is the midpoint of \( HG \). Round \( \varphi \) to the nearest degree. Help: Choose the appropriate coordinate system.
\( \varphi\doteq 104^{\circ} \)
\( \varphi\doteq 76^{\circ} \)
\( \varphi\doteq 100^{\circ} \)
\( \varphi\doteq 80^{\circ} \)