Exam-Style Questions.

Consider two perpendicular vectors \(p\) and \(q\).

(a) Let \(r=p-q\). Draw a diagram to show what this relationship might look like.

(b) If \(p=\begin{pmatrix} 4 \\ 1 \\ -3 \\ \end{pmatrix}\) and \(q=\begin{pmatrix} 3 \\ n \\ -5 \\ \end{pmatrix}\), where \(n\in \mathbb Z\), find \(n\).

Worked Solution

The line \(L_1\) passes through the points A(3, 5, 1) and B(3, 6, 0).

(a) Show that \(AB=\begin{pmatrix} 0 \\ 1 \\ -1 \\ \end{pmatrix}\)

(b) Find a direction vector for \(L_1\)

(c) a vector equation for \(L_1\)

Another line \(L_2\) has equation \(\begin{pmatrix} 8 \\ 3 \\ -2 \\ \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 0 \\ \end{pmatrix}\). The lines \(L_1\) and \(L_2\) intersect at the point P.

(d) Find the coordinates of P.

(e) Find a direction vector for \(L_2\).

(f) Hence, find the angle between \(L_1\) and \(L_2\).

Worked Solution

Two points \(A\) and \(B\) have coordinates (1 , 3 , 6) and (8 , 7 , 10) respectively.

(a) Find \( \overrightarrow{AB} \) in terms of the unit vectors \(i, j\) and \(k\).

(b) Find \(\mid\overrightarrow{AB} \mid\)

Let \( \overrightarrow{AC} = 5i + 2j - k\)

(c) Find the angle between \(AB\) and \(AC\).

(d) Find the area of triangle \(ABC\).

(e) Hence or otherwise find the shortest distance from \(C\) to the line through \(A\) and \(B\).

Worked Solution

Consider the vectors \(\mathbf{a}\) and \(\mathbf{b}\) such that \(\mathbf{a} = \begin{pmatrix} 16 \\ -12 \end{pmatrix} \) and \( |\mathbf{b}| = 11\).

(a) Find the possible range of values for \(|\mathbf{a+b}|\).

Consider the vector \(\mathbf{p}\) such that \(\mathbf{p=a+b}\).

(b) Given that \(|\mathbf{a+b}|\) is a minimum, find \(\mathbf{p}\).

Consider the vector q such that \(\mathbf{q} = \begin{pmatrix}x \\ y \end{pmatrix} \) , where \(x,y \in \mathbb{R} \).

(c) Find \(\mathbf{q}\) such that \(\mathbf{|q| = |b|}\) and \(\mathbf{q}\) is perpendicular to \(\mathbf{a}\).

Worked Solution