Given that \(\mathbf{a}\) and \(\mathbf{b}\) are non-zero vectors such that \(\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\), find the relationship between \(\mathbf{a}\) and \(\mathbf{b}\). Justify your answer.[2]
A solar panel array has been installed on the rooftop of a building. Points \((x, y, z)\) are defined relative to the origin \((0, 0, 0)\) on the horizontal ground modelled by the \(x-y\) plane. The thickness of the solar panel array can be neglected.
The point \(A(2, 3, 4)\), \(B(6, 3, 3)\) and \(C(14, 4, 1)\) lie on the solar panel array modelled by the plane \(\pi\). Find the cartesian equation of \(\pi\).[3]
The solar panel array \(\pi\) at the rooftop is tilted at an acute angle of \(\theta^\circ\) from the horizontal ground. Find \(\theta\).[2]
Mounted above the solar panel array is an inspection device. As it moves, the motion of the inspection device can be modelled by another plane \(\pi_1\) parallel to \(\pi\). The thickness of \(\pi_1\) can be neglected.
Given that the inspection device is \(\frac{1}{\sqrt{17}}\) units from the solar panel array, find the two possible equations of \(\pi_1\) in scalar product form. Give a reason for rejecting one of the equations.[3]
The inspection device is now at point \((3,0,4)\). It emits a ray of light that is perpendicular to \(\pi\). Find the position vector of the point where the ray of light hits \(\pi\).[2]
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Answer:(a) parallel, same direction (b)(i) \(x+4z=18\) (b)(ii) \(14.0^\circ\) (b)(iii) \(\mathbf r\cdot(1,0,4)=19\) (b)(iv) \(\dfrac2{17}(25,0,32)\)