(a) Find, correct to one decimal place, the angle... | WAEC SSCE - Further Mathematics - 2007

Question 1 Report

(a) Find, correct to one decimal place, the angle between \(p = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\) and \(q = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).

(b) ABCD is a square with vertices at A(0, 0), B(2, 0), C(2, 2) and D(0, 2). Forces of magnitude 10 N, 15 N, 20 N and 5 N act along \(\overrightarrow{BA}, \overrightarrow{BC}, \overrightarrow{DC}\) and \(\overrightarrow{AD}\) respectively. Find the (i) magnitude (ii) direction; of the resultant.

Answer

(a) Angle between \(p=\binom{3}{-1}\) and \(q=\binom{3}{4}\).

\[\cos\theta=\frac{p\cdot q}{|p|\,|q|}=\frac{(3)(3)+(-1)(4)}{\sqrt{3^{2}+(-1)^{2}}\,\sqrt{3^{2}+4^{2}}}=\frac{9-4}{\sqrt{10}\cdot 5}=\frac{5}{5\sqrt{10}}=\frac{1}{\sqrt{10}}\]\[\theta=\cos^{-1}(0.3162)=71.6^{\circ}\ (\text{1 d.p.})\]

(b) Forces on square A(0,0), B(2,0), C(2,2), D(0,2).

Write each force as its magnitude times the unit vector of its line of action: