2021HSCQ7Which diagram best shows the curve described by r⃗(t)=−5cos(t) i⃗+5sin(t) j⃗+t k⃗\vec r(t)=-5\cos(t)\,\vec i+5\sin(t)\,\vec j+t\,\vec kr for ?
2020HSCQ3What is the Cartesian equation of the line r=(13)+λ(−24)\mathbf r=\begin{pmatrix}1\\3\end{pmatrix}+\lambda\begin{pmatrix}-2\\4\end{pmatrix}r=(13)+λ(−24)? A. 2y+x=72y+x=72y+x=7 B. y−2x=−5y-2x=-5y−2x=−5 C. y+2x=5y+2x=5y D. 1 markMultiple choice
2020HSCQ13 (b)Consider two lines in three dimensions given by: r=(3−17)+λ1(121)andr=(3−62)+λ2(−213).\mathbf r=\begin{pmatrix}3\\-1\\7\end{pmatrix}+\lambda_1\begin{pmatrix}1\\2\\1\end{pmatrix} \quad \text{and} \quad \mathbf r=\begin{pmatrix}3\\-6\\2\end{pmatrix}+\lambda_2\begin{pmatrix}-2\\1\\3\end{pmatrix}.r=3−17 By equating components, find the point of intersection of the two lines.3 marksShort answer
2020HSCQ15 (b) (iii)Let OPQROPQROPQR be a parallelogram with OP→=p\overrightarrow{OP}=\mathbf pOP=p and OR→=r\overrightarrow{OR}=\mathbf rOR=r. The point SSS is the midpoint of QRQRQR and TTT is the intersection of PRPRPR and OSOSOS, as shown in the diagram. (iii) Show that OT→=23r+13p.\overrightarrow{OT}=\frac23\mathbf r+\frac13\mathbf p.OT=323 marksShort answer