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The diagram shows a cube \(ABCDEFGH\).
Write the following vectors in terms of vectors
\(\mathbf{a}=\overrightarrow{AB},~ \mathbf{b}=\overrightarrow{BC}\) and \(\mathbf{c}=\overrightarrow{CG}\):
(a) \(\overrightarrow{AG}\)
(b) \(\overrightarrow{AH}\)
(c) \(\overrightarrow{CE}\)
(d) \(\overrightarrow{GD}\)
Solutions:
(a) \(\overrightarrow{AG}=\mathbf{a}+\mathbf{b}+\mathbf{c}\);
(b) \(\overrightarrow{AH}=\mathbf{b}+\mathbf{c}\);
(c) \(\overrightarrow{CE}=-\mathbf{a}-\mathbf{b}+\mathbf{c}\);
(d) \(\overrightarrow{GD}=-\mathbf{a}-\mathbf{c}\)
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The points \(A(0,0,0),~ B(3,0,0),~ D(0,7,0)\) and \(E(0,0,5)\) are vertices of the cuboid \(ABCDEFGH\).
Express the following vectors in terms of the standard basis vectors \(\mathbf{i},~\mathbf{j}\) and
\(\mathbf{k}\):
(a) \(\overrightarrow{AC}\)
(b) \(\overrightarrow{AG}\)
(c) \(\overrightarrow{EC}\)
(d) \(\overrightarrow{HB}\)
Solutions:
(a) \(\overrightarrow{AC}=3\mathbf{i}+7\mathbf{j}\);
(b) \(\overrightarrow{AG}=3\mathbf{i}+7\mathbf{j}+5\mathbf{k}\);
(c) \(\overrightarrow{EC}=3\mathbf{i}+7\mathbf{j}-5\mathbf{k}\);
(d) \(\overrightarrow{HB}=3\mathbf{i}-7\mathbf{j}-5\mathbf{k}\)
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Vectors have the coordinates:
\(\mathbf{a}=\left(\begin{array}{c} 4 \\ 1 \\ 2 \end{array}\right)\),
\(\mathbf{b}=\left(\begin{array}{c} 3 \\ 5 \\ 1 \end{array}\right)\),
\(\mathbf{c}=\left(\begin{array}{c} 6 \\ 2 \\ -3 \end{array}\right)\).
Calculate vectors:
(a) \(\mathbf{u}=\mathbf{a}+\mathbf{b}+\mathbf{c}\)
(b) \(\mathbf{v}=\mathbf{a}-\mathbf{b}-\mathbf{c}\)
(c) \(\mathbf{w}=2\mathbf{a}+3\mathbf{b}-\mathbf{c}\)
Solutions:
(a) \(\mathbf{u}=\left(\begin{array}{c} 13 \\ 8 \\ 0 \end{array}\right)\);
(b) \(\mathbf{v}=\left(\begin{array}{c} -5 \\ -6 \\ 4 \end{array}\right)\);
(c) \(\mathbf{w}=\left(\begin{array}{c} 11 \\ 15 \\ 10 \end{array}\right)\)
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Let
\(\mathbf{a}=8\mathbf{i}+7\mathbf{j}+4\mathbf{k}\) and
\(\mathbf{b}=5\mathbf{i}+\mathbf{j}+6\mathbf{k}\).
Calculate:
(a) \(\mathbf{a}+\mathbf{b}\)
(b) \(\mathbf{a}-\mathbf{b}\)
(c) \(|\mathbf{a}-\mathbf{b}|\)
Solutions:
(a) \(\mathbf{a}+\mathbf{b}=13\mathbf{i}+8\mathbf{j}+10\mathbf{k}\);
(b) \(\mathbf{a}-\mathbf{b}=3\mathbf{i}+6\mathbf{j}-2\mathbf{k}\);
(c) \(|\mathbf{a}-\mathbf{b}|=7\)
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Let
\(\mathbf{a}=\mathbf{i}-\mathbf{j}-\mathbf{k}\),
\(\mathbf{b}=\mathbf{j}-\mathbf{i}-\mathbf{k}\) and
\(\mathbf{c}=\mathbf{k}-\mathbf{j}\).
Calculate:
(a) \(\mathbf{a}+\mathbf{b}+\mathbf{c}\)
(b) \(\mathbf{b}-2\mathbf{a}-\mathbf{c}\)
Solutions:
(a) \(\mathbf{a}+\mathbf{b}+\mathbf{c}=-\mathbf{j}-\mathbf{k}\);
(b) \(\mathbf{b}-2\mathbf{a}-\mathbf{c}=-3\mathbf{i}+4\mathbf{j}\)
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Given vector \(\mathbf{a}=\left(\begin{array}{c} 12 \\ 8 \\ 9 \end{array}\right)\)
(a) find \(|\mathbf{a}|\),
(b) find the unit vector \(\mathbf{u}\) which
has the same direction as \(\mathbf{a}\).
Solutions:
(a) \(|\mathbf{a}|=17\);
(b) \(\mathbf{u}=\left(\begin{array}{c} \frac{12}{17} \\ \frac{8}{17} \\ \frac{9}{17} \end{array}\right)\)
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Let \(\mathbf{a}=4\mathbf{i}+\mathbf{j}-8\mathbf{k}\).
Find the unit vector in the direction of the vector \(\mathbf{a}\).
Solutions:
\(\mathbf{u}=\frac{4}{9}\mathbf{i}+\frac{1}{9}\mathbf{j}-\frac{8}{9}\mathbf{k}\)
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Points \(P\) and \(Q\) have the position vectors
\(\overrightarrow{OP}=5\mathbf{i}-\mathbf{j}+2\mathbf{k}\) and
\(\overrightarrow{OQ}=7\mathbf{i}+5\mathbf{j}-\mathbf{k}\).
(a) Write down vector \(\overrightarrow{PQ}\).
(b) Find \(|\overrightarrow{PQ}|\).
Solutions:
(a) \(\overrightarrow{PQ}=2\mathbf{i}+6\mathbf{j}-3\mathbf{k}\);
(b) \(|\overrightarrow{PQ}|=7\)
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Points \(B(5,2,3),~ C(7,6,2)\) and \(D(1,8,1)\) are three of the vertices of the parallelogram \(ABCD\).
(a) Find the coordinates of the point \(A\).
(b) Find the coordinates of the intersection point of the diagonals.
Solutions:
(a) \(A(-1,4,2)\);
(b) \(P(3,5,2)\)
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Points \(A(-2,1,1)\) and \(D(1,3,2)\) are two of the vertices of the parallelogram \(ABCD\).
Diagonals of this parallelogram intersect at \(P(7,-1,3)\).
Find the coordinates of points \(B\) and \(C\).
Solutions:
\(B(13,-5,4),~ C(16,-3,5)\)
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Let \(\mathbf{v}=\left(\begin{array}{c} 6 \\ 4 \\ 8 \end{array}\right)\).
Find out which of the following vectors are parallel to \(\mathbf{v}\):
\(\mathbf{a}=\left(\begin{array}{c} -12 \\ -8 \\ -16 \end{array}\right)\),
\(\mathbf{b}=\left(\begin{array}{c} 24 \\ 16 \\ 30 \end{array}\right)\),
\(\mathbf{c}=\left(\begin{array}{c} 9 \\ 6 \\ 12 \end{array}\right)\)
Solutions:
\(\mathbf{a}\) is parallel to \(\mathbf{v}\),
\(\mathbf{b}\) is not parallel to \(\mathbf{v}\),
\(\mathbf{c}\) is parallel to \(\mathbf{v}\)
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Let
\(\mathbf{v}=\left(\begin{array}{c} 5 \\ -7 \\ 5 \end{array}\right)\),
\(\mathbf{a}=\left(\begin{array}{c} 1 \\ 0 \\ 2 \end{array}\right)\),
\(\mathbf{b}=\left(\begin{array}{c} 2 \\ 1 \\ 0 \end{array}\right)\),
\(\mathbf{c}=\left(\begin{array}{c} 0 \\ 3 \\ -1 \end{array}\right)\).
Express vector \(\mathbf{v}\) in terms of \(\mathbf{a},~ \mathbf{b}\) and \(\mathbf{c}\).
Solutions:
\(\mathbf{v}=\mathbf{a}+2\mathbf{b}-3\mathbf{c}\)
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Let
\(\mathbf{a}=\left(\begin{array}{c} 1 \\ 2 \\ 5 \end{array}\right)\),
\(\mathbf{b}=\left(\begin{array}{c} 7 \\ -1 \\ 3 \end{array}\right)\),
\(\mathbf{c}=\left(\begin{array}{c} 2 \\ 6 \\ 0 \end{array}\right)\),
\(\mathbf{d}=\left(\begin{array}{c} -4 \\ 8 \\ 7 \end{array}\right)\).
Express vector \(\mathbf{d}\) in terms of \(\mathbf{a},~ \mathbf{b}\) and \(\mathbf{c}\).
Solutions:
\(\mathbf{d}=2\mathbf{a}-\mathbf{b}+\frac{1}{2}\mathbf{c}\)
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Let
\(\mathbf{a}=\left(\begin{array}{c} 1 \\ 1 \\ 3 \end{array}\right)\),
\(\mathbf{b}=\left(\begin{array}{c} 0 \\ 2 \\ 2 \end{array}\right)\),
\(\mathbf{c}=\left(\begin{array}{c} 3 \\ 7 \\ 13 \end{array}\right)\).
Express vector \(\mathbf{c}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) if possible.
Solutions:
It is possible: \(\mathbf{c}=3\mathbf{a}+2\mathbf{b}\)
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Let
\(\mathbf{a}=\left(\begin{array}{c} 2 \\ 1 \\ -3 \end{array}\right)\),
\(\mathbf{b}=\left(\begin{array}{c} 1 \\ 3 \\ -1 \end{array}\right)\),
\(\mathbf{c}=\left(\begin{array}{c} 3 \\ -1 \\ 2 \end{array}\right)\).
Express vector \(\mathbf{c}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) if possible.
Solutions:
It is not possible.
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Line segment \(AB\) has the endpoints \(A(-1,3,2)\) and \(B(14,12,-1)\). Find the coordinates of the points
\(P\) and \(Q\) which divide \(AB\) in three equal parts.
Solutions:
\(P(4,6,1),~ Q(9,9,0)\)
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Points \(U(3,5,-2)\) and \(V(8,-5,8)\) are the endpoints of the line segment \(UV\).
Points \(E,~ F,~G\) and \(H\) divide \(UV\) in five equal parts.
Find the coordinates of the points \(E,~ F,~G\) and \(H\).
Solutions:
\(E(4,3,0),~ F(5,1,2),~ G(6,-1,4),~ H(7,-3,6)\)
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Points \(A(3,2,-1)\) and \(B(17,-5,6)\) are the endpoints of the line segment \(AB\) and
\(P\) is the point on \(AB\) such that \(AP:PB=3:4\).
Find the coordinates of the point \(P\).
Solutions:
\(P(9,-1,2)\)
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Vector \(\mathbf{a}=\left(\begin{array}{c} 9 \\ 2 \\ z \end{array}\right)\) has the modulus
\(|\mathbf{a}|=11\). Find the unknown coordinate \(z\). Write down all possible solutions.
Solutions:
\(z_1=6,~ z_2=-6\)
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Vector \(\mathbf{v}=\mathbf{i}+m\mathbf{j}+2m\mathbf{k}\) has the modulus
\(|\mathbf{v}|=9\). Find the value of \(m\). Write down all possible solutions.
Solutions:
\(m_1=4,~ m_2=-4\)
