Points and Vectors

1103024309

Level: 
A
Given the vectors \( \vec{a} \), \( \vec{b} \), \( \vec{c} \) shown in the picture, express a vector \( \vec{b} \) as a linear combination of vectors \( \vec{a} \) and \( \vec{c} \).
\( \vec{b} = 2\vec{a} + \vec{c} \)
\( \vec{b} = 2\vec{a} - \vec{c} \)
\( \vec{b} = -2\vec{a} + \vec{c} \)
\( \vec{b} = -2\vec{a} - \vec{c} \)

1103024308

Level: 
A
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 \( \vec{b} \).
\( \vec{c} = -2\vec{a} + \vec{b} \)
\( \vec{c} = -\vec{a} + \frac12\vec{b} \)
\( \vec{c} = -\frac32\vec{a} + \vec{b} \)
\( \vec{c} = -2\vec{a} + \frac32\vec{b} \)

1003024307

Level: 
A
Let \( \vec{a} = (-1;2) \), \( \vec{b} = (2;1) \), \( \vec{c} = (-4;3) \). Express vector \( \vec{c} \) as a linear combination of vectors \( \vec{a} \) and \( \vec{b} \).
\( \vec{c} = 2\vec{a} - \vec{b} \)
\( \vec{c} = 4\vec{a} - 8\vec{b} \)
\( \vec{c} = 4\vec{a} - \vec{b} \)
\( \vec{c} = -2\vec{a} + \vec{b} \)

1003024306

Level: 
A
We are given the points A = [-4;2;3], B = [-5;6;3], D = [1;1;4]. Find the coordinates of a point \( C \), if: \[ \vec{u} = \overrightarrow{AB}\text{, }\ \overrightarrow{CD} = -\frac12\vec{u}\]
\( C = \left[\frac12; 3; 4\right] \)
\( C = \left[-\frac12;-3;-4\right] \)
\( C = \left[\frac32;3;4\right] \)
\( C = \left[\frac32;-3;-4\right] \)

1103024305

Level: 
A
In a tetrahedron \( ABCD \), let \( \vec{b} = \overrightarrow{AB} \), \( \vec{c} = \overrightarrow{AC} \), \( \vec{d} = \overrightarrow{AD} \), \( \vec{e} = \overrightarrow{AE} \) and \( \vec{f} = \overrightarrow{DE} \). Further let \( E \) be the midpoint of \( BC \). Express vectors \( \vec{e} \) and \( \vec{f} \) as a linear combination of vectors \( \vec{b} \), \( \vec{c} \), \( \vec{d} \).
\( \vec{e} = \frac12\vec{b} + \frac12\vec{c};\ \vec{f} = \frac12\vec{b} + \frac12\vec{c} - \vec{d} \)
\( \vec{e} = \frac12\vec{b} + \frac12\vec{d};\ \vec{f} = \vec{b} + \vec{c} + \vec{d} \)
\( \vec{e} = \vec{b} + \vec{c};\ \vec{f} =\frac12\vec{b} + \frac12\vec{c} - \vec{d} \)
\( \vec{e} = \frac12\vec{b} + \frac12\vec{c};\ \vec{f} = \frac12\vec{b} + \frac12\vec{c} + \vec{d} \)

1103024304

Level: 
A
The picture shows a rectangular cuboid \( ABCDEFGH \). In the cuboid find the vector that is the sum of \( \overrightarrow{BC} + \overrightarrow{AE} + \overrightarrow{CF} + \overrightarrow{FA} + \overrightarrow{HG} \).
\( \overrightarrow{BF} \)
\( \overrightarrow{BE} \)
\( \overrightarrow{BG} \)
\( \overrightarrow{BH} \)

1103024303

Level: 
A
The picture shows a rectangular cuboid \( ABCDEFGH \) with \( \vec{a} = \overrightarrow{AB} \), \( \vec{b} = \overrightarrow{AD} \), \( \vec{c} = \overrightarrow{AE} \), \( \vec{x} = \overrightarrow{AK} \) and \( \vec{y} = \overrightarrow{AL} \). Point \( K \) is the midpoint of \( FG \) and point \( L \) is the centre of face \( BCGF \). Express vectors \( \vec{x} \) and \( \vec{y} \) as a linear combination of vectors \( \vec{a} \), \( \vec{b} \), \( \vec{c} \).
\( \vec{x} = \vec{a} + \frac12\vec{b} + \vec{c};\ \vec{y} = \vec{a} + \frac12\vec{b} + \frac12\vec{c} \)
\( \vec{x} = \frac12\vec{a} + \vec{b} + \frac12\vec{c};\ \vec{y} = \vec{a} - \frac12\vec{b} + \frac12\vec{c} \)
\( \vec{x} = \vec{a} + \frac12\vec{b} + \frac12\vec{c};\ \vec{y} = \vec{a} - \frac12\vec{b} + \frac12\vec{c} \)
\( \vec{x} = \vec{a} + \frac12\vec{b} + \frac12\vec{c};\ \vec{y} = \frac12\vec{a} + \frac12\vec{b} + \frac12\vec{c} \)

1103024302

Level: 
A
In a regular hexagon \( ABCDEF \) shown in the picture, let \( \vec{a} = \overrightarrow{AB} \), \( \vec{b} = \overrightarrow{BC} \), \( \vec{c} = \overrightarrow{FD} \) and \( \vec{d} = \overrightarrow{CD} \). Express vectors \( \vec{c} \) and \( \vec{d} \) as a linear combination of vectors \( \vec{a} \) and \( \vec{b} \).
\( \vec{c} = \vec{a} + \vec{b};\ \vec{d} = \vec{b} - \vec{a} \)
\( \vec{c} = 2\vec{a} + 2\vec{b};\ \vec{d} = 2\vec{b} - 0.5\vec{a} \)
\( \vec{c} = 2\vec{a} + \vec{b};\ \vec{d} = \vec{b} - \vec{a} \)
\( \vec{c} = \vec{a} + \vec{b};\ \vec{d} = \vec{a} - \vec{b} \)

1103024301

Level: 
A
In a triangle \( ABC \), let \( K \), \( L \) and \( M \) be the midpoints of \( AB \), \( BC \) and \( AC \) consecutively and let \( T \) be the centroid of \( ABC \). Find the values of coefficients \( k \), \( l \) and \(m \) if: \[ \overrightarrow{TM} = k\cdot\overrightarrow{BT};\ \overrightarrow{ML} = l\cdot\overrightarrow{BA};\ \overrightarrow{CK} = m\cdot\overrightarrow{TC} \]
\( k=\frac12;\ l=-\frac12 ;\ m=-\frac32 \)
\( k=\frac12;\ l=\frac12;\ m=-\frac32 \)
\( k=\frac12 ;\ l=-\frac12 ;\ m=-\frac23 \)
\( k=\frac12;\ l=-\frac12;\ m=\frac32 \)

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 \( \vec{v}\cdot\vec{u} \).
\( \vec{v}\cdot\vec{w}=9 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=27 \)
\( \vec{v}\cdot\vec{w}=9 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=9\sqrt6 \)
\( \vec{v}\cdot\vec{w}=\frac92 \), \( \vec{v}\cdot\vec{z} = 0 \), \( \vec{v}\cdot\vec{u}=9\sqrt6 \)
\( \vec{v}\cdot\vec{w}=\frac92 \), \( \vec{v}\cdot\vec{z} = 1 \), \( \vec{v}\cdot\vec{u}=27 \)