Relative to an origin O, the position vectors of three points, A, B and C, are given by
\(\overrightarrow{OA} = \mathbf{i} + 2p\mathbf{j} + q\mathbf{k}, \quad \overrightarrow{OB} = q\mathbf{j} - 2p\mathbf{k} \quad \text{and} \quad \overrightarrow{OC} = -(4p^2 + q^2)\mathbf{i} + 2p\mathbf{j} + q\mathbf{k},\)
where \(p\) and \(q\) are constants.
The position vectors of points A and B relative to an origin O are given by
\(\overrightarrow{OA} = \begin{pmatrix} p \\ 1 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ p \end{pmatrix}\),
where \(p\) is a constant.
The position vectors of the points A and B, relative to an origin O, are given by
\(\overrightarrow{OA} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} k \\ -k \\ 2k \end{pmatrix}\),
where \(k\) is a constant.
The position vectors of points A and B relative to an origin O are a and b respectively. The position vectors of points C and D relative to O are 3a and 2b respectively. It is given that
\(\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 4 \\ 0 \\ 6 \end{pmatrix}\).
(i) Find the unit vector in the direction of \(\overrightarrow{CD}\).
(ii) The point E is the mid-point of CD. Find angle EOD.
Relative to an origin O, the position vectors of the points A, B and C are given by
\(\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 1 \\ 3 \\ p \end{pmatrix}.\)
Find
(i) the unit vector in the direction of \(\overrightarrow{AB}\),
(ii) the value of the constant \(p\) for which angle \(BOC = 90^\circ\).
(i) Find the angle between the vectors \(3\mathbf{i} - 4\mathbf{k}\) and \(2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}\).
The vector \(\overrightarrow{OA}\) has a magnitude of 15 units and is in the same direction as the vector \(3\mathbf{i} - 4\mathbf{k}\). The vector \(\overrightarrow{OB}\) has a magnitude of 14 units and is in the same direction as the vector \(2\mathbf{i} + 3\mathbf{j} - 6\mathbf{k}\).
(ii) Express \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) in terms of \(\mathbf{i}, \mathbf{j}\) and \(\mathbf{k}\).
(iii) Find the unit vector in the direction of \(\overrightarrow{AB}\).
Two vectors u and v are such that u = \(\begin{pmatrix} p^2 \\ -2 \\ 6 \end{pmatrix}\) and v = \(\begin{pmatrix} 2 \\ p-1 \\ 2p+1 \end{pmatrix}\), where \(p\) is a constant.
(i) Find the values of \(p\) for which u is perpendicular to v.
(ii) For the case where \(p = 1\), find the angle between the directions of u and v.
Relative to an origin O, the position vectors of points A and B are 3i + 4j - k and 5i - 2j - 3k respectively.
(i) Use a scalar product to find angle BOA.
The point C is the mid-point of AB. The point D is such that \(\overrightarrow{OD} = 2\overrightarrow{OB}\).
(ii) Find \(\overrightarrow{DC}\).
Relative to an origin O, the position vectors of points A and B are given by \(\overrightarrow{OA} = 5\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) and \(\overrightarrow{OB} = 2\mathbf{i} + 7\mathbf{j} + p\mathbf{k}\), where \(p\) is a constant.
(i) Find the value of \(p\) for which angle \(AOB\) is \(90^\circ\).
(ii) In the case where \(p = 4\), find the vector which has magnitude 28 and is in the same direction as \(\overrightarrow{AB}\).
Relative to an origin O, the point A has position vector \(4\mathbf{i} + 7\mathbf{j} - p\mathbf{k}\) and the point B has position vector \(8\mathbf{i} - \mathbf{j} - p\mathbf{k}\), where \(p\) is a constant.
Relative to the origin \(O\), the points \(A, B\) and \(D\) have position vectors given by
\(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + \mathbf{k}, \quad \overrightarrow{OB} = 2\mathbf{i} + 5\mathbf{j} + 3\mathbf{k} \quad \text{and} \quad \overrightarrow{OD} = 3\mathbf{i} + 2\mathbf{k}.\)
A fourth point \(C\) is such that \(ABCD\) is a parallelogram.
(a) Find the position vector of \(C\) and verify that the parallelogram is not a rhombus. [5]
(b) Find angle \(BAD\), giving your answer in degrees. [3]
(c) Find the area of the parallelogram correct to 3 significant figures. [2]
Relative to the origin O, the position vectors of the points A, B and C are given by
\(\overrightarrow{OA} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 10 \\ 0 \\ 6 \end{pmatrix}.\)
(i) Find angle \(ABC\).
The point D is such that ABCD is a parallelogram.
(ii) Find the position vector of D.
Relative to an origin O, the position vectors of the points A, B and C are given by
\(\overrightarrow{OA} = i - 2j + 4k, \quad \overrightarrow{OB} = 3i + 2j + 8k, \quad \overrightarrow{OC} = -i - 2j + 10k.\)
Relative to an origin O, the position vectors of the points A and B are given by
\(\overrightarrow{OA} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 4 \\ 1 \\ p \end{pmatrix}\).
(i) Find the value of p for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\).
(ii) Find the values of p for which the magnitude of \(\overrightarrow{AB}\) is 7.
Relative to an origin O, the position vectors of the points A, B and C are given by
\(\overrightarrow{OA} = \begin{pmatrix} 2 \\ 3 \\ -6 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 0 \\ -6 \\ 8 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} -2 \\ 5 \\ -2 \end{pmatrix}.\)
Relative to an origin O, the position vectors of the points A and B are given by \(\overrightarrow{OA} = 2\mathbf{i} - 8\mathbf{j} + 4\mathbf{k}\) and \(\overrightarrow{OB} = 7\mathbf{i} + 2\mathbf{j} - \mathbf{k}\).
(i) Find the value of \(\overrightarrow{OA} \cdot \overrightarrow{OB}\) and hence state whether angle AOB is acute, obtuse or a right angle.
(ii) The point X is such that \(\overrightarrow{AX} = \frac{2}{5} \overrightarrow{AB}\). Find the unit vector in the direction of \(\overrightarrow{OX}\).
Relative to an origin O, the position vectors of points A and B are \(2\mathbf{i} + \mathbf{j} + 2\mathbf{k}\) and \(3\mathbf{i} - 2\mathbf{j} + p\mathbf{k}\) respectively.
Relative to an origin O, the position vectors of the points A and B are given by
\(\overrightarrow{OA} = \begin{pmatrix} 4 \\ 1 \\ -2 \end{pmatrix}\) and \(\overrightarrow{OB} = \begin{pmatrix} 3 \\ 2 \\ -4 \end{pmatrix}\).
(i) Given that C is the point such that \(\overrightarrow{AC} = 2\overrightarrow{AB}\), find the unit vector in the direction of \(\overrightarrow{OC}\).
The position vector of the point D is given by \(\overrightarrow{OD} = \begin{pmatrix} 1 \\ 4 \\ k \end{pmatrix}\), where k is a constant, and it is given that \(\overrightarrow{OD} = m\overrightarrow{OA} + n\overrightarrow{OB}\), where m and n are constants.
(ii) Find the values of m, n and k.
The position vectors of points A and B are \(\begin{pmatrix} -3 \\ 6 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 2 \\ 4 \end{pmatrix}\) respectively, relative to an origin O.
(i) Calculate angle \(AOB\).
(ii) The point C is such that \(\overrightarrow{AC} = 3\overrightarrow{AB}\). Find the unit vector in the direction of \(\overrightarrow{OC}\).
Relative to an origin O, the position vectors of points P and Q are given by
\(\overrightarrow{OP} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OQ} = \begin{pmatrix} 2 \\ 1 \\ q \end{pmatrix}\),
where \(q\) is a constant.