Points and Vectors

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: \[ \overrightarrow{TM} = k\cdot\overrightarrow{BT};\ \overrightarrow{ML} = l\cdot\overrightarrow{BA};\ \overrightarrow{CK} = m\cdot\overrightarrow{TC} \]
\( k=\frac12;\ l=-\frac12 ;\ m=-\frac32 \)
\( k=\frac12;\ l=\frac12;\ m=-\frac32 \)
\( k=\frac12 ;\ l=-\frac12 ;\ m=-\frac23 \)
\( k=\frac12;\ l=-\frac12;\ m=\frac32 \)

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 \( \vec{b} \).
\( \vec{c} = \vec{a} + \vec{b};\ \vec{d} = \vec{b} - \vec{a} \)
\( \vec{c} = 2\vec{a} + 2\vec{b};\ \vec{d} = 2\vec{b} - 0.5\vec{a} \)
\( \vec{c} = 2\vec{a} + \vec{b};\ \vec{d} = \vec{b} - \vec{a} \)
\( \vec{c} = \vec{a} + \vec{b};\ \vec{d} = \vec{a} - \vec{b} \)

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 \( \vec{a} \), \( \vec{b} \), \( \vec{c} \).
\( \vec{x} = \vec{a} + \frac12\vec{b} + \vec{c};\ \vec{y} = \vec{a} + \frac12\vec{b} + \frac12\vec{c} \)
\( \vec{x} = \frac12\vec{a} + \vec{b} + \frac12\vec{c};\ \vec{y} = \vec{a} - \frac12\vec{b} + \frac12\vec{c} \)
\( \vec{x} = \vec{a} + \frac12\vec{b} + \frac12\vec{c};\ \vec{y} = \vec{a} - \frac12\vec{b} + \frac12\vec{c} \)
\( \vec{x} = \vec{a} + \frac12\vec{b} + \frac12\vec{c};\ \vec{y} = \frac12\vec{a} + \frac12\vec{b} + \frac12\vec{c} \)

1103024304

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

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 \( \vec{b} \), \( \vec{c} \), \( \vec{d} \).
\( \vec{e} = \frac12\vec{b} + \frac12\vec{c};\ \vec{f} = \frac12\vec{b} + \frac12\vec{c} - \vec{d} \)
\( \vec{e} = \frac12\vec{b} + \frac12\vec{d};\ \vec{f} = \vec{b} + \vec{c} + \vec{d} \)
\( \vec{e} = \vec{b} + \vec{c};\ \vec{f} =\frac12\vec{b} + \frac12\vec{c} - \vec{d} \)
\( \vec{e} = \frac12\vec{b} + \frac12\vec{c};\ \vec{f} = \frac12\vec{b} + \frac12\vec{c} + \vec{d} \)

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

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 \( \vec{c} \).
\( \vec{b} = 2\vec{a} + \vec{c} \)
\( \vec{b} = 2\vec{a} - \vec{c} \)
\( \vec{b} = -2\vec{a} + \vec{c} \)
\( \vec{b} = -2\vec{a} - \vec{c} \)

1103024310

Level: 
A
The picture shows the triangle \( KLM \) with indicated vectors \( \vec{a} \), \( \vec{b} \), \( \vec{c} \) in a coordinate system. What are the vector coordinates \( \vec{b} \)? Express \( \vec{b} \) as a linear combination of \( \vec{a} \) and \( \vec{c} \).
\( \vec{b} = \left(1;3;4.5\right);\ \vec{b} = \frac12\vec{a} + \frac12\vec{c} \)
\( \vec{b} = \left(3;1;4.5\right);\ \vec{b} = \vec{a} + \vec{c} \)
\( \vec{b} = \left(1;3;4.5\right);\ \vec{b} = \vec{a} + \vec{c} \)
\( \vec{b} = \left(3;1;4.5\right);\ \vec{b} = \frac12\vec{a} + \frac12\vec{c} \)

1103030701

Level: 
A
We are given points \( A = [1;-1;2] \), \( B = [0;5;-3] \), \( S = [2;0;5] \). Point \( S \) is the centre of a parallelogram \( ABCD \). Find the coordinates of vertices \( C \) and \( D \).
\( C = [3;1;8]; D = [4;-5;13] \)
\( C = [4;-5;13]; D = [3;1;8] \)
\( C = [1;1;3]; D = [2;-5;8] \)
\( C = [-3;-1;-8]; D = [-4;5;-13] \)