1.10a Vectors in 2D: i,j notation and column vectors

The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by $$\mathbf{a} = -3\mathbf{i} + 4\mathbf{j}, \quad \mathbf{b} = 5\mathbf{i} + 8\mathbf{j},$$ respectively.

1. Find the vector \(\overrightarrow{AB}\). [2]
2. 1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
   2. The point \(C\) is such that the vector \(\overrightarrow{OC}\) is in the direction of \(\mathbf{a}\). Given that the length of \(\overrightarrow{OC}\) is 7 units, write down the position vector of \(C\). [1]
3. Calculate the angle \(AOB\). [3]

Vector \(\mathbf{v} = a\mathbf{i} + 0.6\mathbf{j}\), where \(a\) is a constant.

1. Given that the direction of \(\mathbf{v}\) is \(45°\), state the value of \(a\). [1]
2. Given instead that \(\mathbf{v}\) is parallel to \(8\mathbf{i} + 3\mathbf{j}\), find the value of \(a\). [2]
3. Given instead that \(\mathbf{v}\) is a unit vector, find the possible values of \(a\). [3]

A particle \(P\) is acted upon by three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) given by \(\mathbf{F}_1 = (6\mathbf{i} - 4\mathbf{j}) \text{ N}\), \(\mathbf{F}_2 = (-3\mathbf{i} + 9\mathbf{j}) \text{ N}\) and \(\mathbf{F}_3 = (a\mathbf{i} + b\mathbf{j}) \text{ N}\), where \(a\) and \(b\) are constants. Given that \(P\) is in equilibrium,

1. find the value of \(a\) and the value of \(b\). [2]

The force \(\mathbf{F}_3\) is now removed. The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\).

1. Find the magnitude of \(\mathbf{R}\). [3]
2. Find the angle, to \(0.1°\), that \(\mathbf{R}\) makes with \(\mathbf{i}\). [2]

Relative to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(4\mathbf{i} + 2\mathbf{j}\), \(3\mathbf{i} + 4\mathbf{j}\) and \(-\mathbf{i} + 12\mathbf{j}\), respectively.

1. Find the magnitude of the vector \(\overrightarrow{OC}\) [2]
2. Find the angle that the vector \(\overrightarrow{OB}\) makes with the vector \(\mathbf{j}\) to the nearest degree [2]
3. Show that the points \(A\), \(B\) and \(C\) are collinear [3]

The points \(A\), \(B\) and \(C\) have position vectors \(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\begin{pmatrix} 6 \\ 3 \end{pmatrix}\) respectively. \(M\) is the midpoint of \(BC\).

1. Find the position vector of the point \(D\) such that \(\overrightarrow{BC} = \overrightarrow{AD}\). [3]
2. Find the magnitude of \(\overrightarrow{AM}\). [3]

The square-based pyramid \(P\) has vertices \(A, B, C, D\) and \(E\). The position vectors of \(A, B, C\) and \(D\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) respectively where $$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$

1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), \(\overrightarrow{AD}\), \(\overrightarrow{BC}\), \(\overrightarrow{BD}\) and \(\overrightarrow{CD}\). [3]
2. Find
   1. the length of a side of the square base of \(P\),
   2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
   3. the height of \(P\),
   4. the position vector of \(E\).
   [9] A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
3. Find the position vector of the other vertex of this octahedron. [3]