(a) Find the angle between the vectors \(a = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\) and \(b = \begin{pmatrix} -8 \\ -15 \end{pmatrix}\). (b) Given that \(a...
Question 1 Report
(a) Find the angle between the vectors \(a = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\) and \(b = \begin{pmatrix} -8 \\ -15 \end{pmatrix}\).
(b) Given that \(a = (4N, 060°)\) and \(b = (3N, 120°)\), find, in component form, the unit vector along \(a - b\).
Answer Details
(a) With \(a = \begin{pmatrix}-3\\4\end{pmatrix},\ b = \begin{pmatrix}-8\\-15\end{pmatrix}\):
\[a\cdot b = (-3)(-8) + (4)(-15) = 24 - 60 = -36\] \[|a| = \sqrt{9+16} = 5,\qquad |b| = \sqrt{64+225} = \sqrt{289} = 17\] \[\cos\theta = \frac{a\cdot b}{|a||b|} = \frac{-36}{85} = -0.4235\] \[\theta = 115.1^{o}\ (\text{to 1 d.p.})\](b) \(a = (4\cos60^{o},\ 4\sin60^{o}) = (2,\ 2\sqrt3)\); \(b = (3\cos120^{o},\ 3\sin120^{o}) = (-1.5,\ 1.5\sqrt3)\).
\[a - b = (2 - (-1.5),\ 2\sqrt3 - 1.5\sqrt3) = (3.5,\ 0.5\sqrt3)\] \[|a - b| = \sqrt{3.5^2 + (0.5\sqrt3)^2} = \sqrt{12.25 + 0.75} = \sqrt{13}\]Unit vector along \(a - b\):
\[\hat u = \frac{1}{\sqrt{13}}\begin{pmatrix} 3.5 \\ 0.5\sqrt3 \end{pmatrix} \approx \begin{pmatrix} 0.971 \\ 0.240 \end{pmatrix}\]Download The App On Google Playstore
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