A

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 \( \vec{c} \).
\( \vec{b} = \left(1,3,4.5\right),\ \vec{b} = \frac12\vec{a} + \frac12\vec{c} \)
\( \vec{b} = \left(3,1,4.5\right),\ \vec{b} = \vec{a} + \vec{c} \)
\( \vec{b} = \left(1,3,4.5\right),\ \vec{b} = \vec{a} + \vec{c} \)
\( \vec{b} = \left(3,1,4.5\right),\ \vec{b} = \frac12\vec{a} + \frac12\vec{c} \)

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