Plane geometry

9000107505

Level: 
B
Find \(\cos \varphi \) where \(\varphi \) is the angle between the lines \(p\) and \(q\). \[ \begin{aligned}[t] p\colon x& = 1 + 4t, & \\y & = 3 - 3t;\ t\in \mathbb{R}; \\ \end{aligned} \quad q\colon x + y - 3 = 0 \]
\(\frac{7\sqrt{2}} {10} \)
\(- \frac{7} {5\sqrt{2}}\)
\(\frac{\sqrt{2}} {5} \)
\(\frac{\sqrt{2}} {10} \)

9000107507

Level: 
B
Find \(\mathop{\mathrm{tg}}\nolimits \varphi \) where \(\varphi \) is the angle between the lines \(p\) and \(q\). \[ \begin{aligned}[t] p\colon x& = 1 + t, & \\y & = 3 + 2t;\ t\in \mathbb{R}; \\ \end{aligned}\quad q\colon y = 1 \]
\(2\)
\(\frac{1} {2}\)
\(- 1\)
\(0\)

9000107509

Level: 
B
In the following list identify a parametric line such that the angle between this line and the line \(q\) is \(0^{\circ }\). \[ q\colon x - 2y + 11 = 0 \]
\(\begin{aligned}[t] p\colon x& = 1 + 4t, & \\y & = 3 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = 1 + 2t, & \\y & = 2 - t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = 2 - t, & \\y & = 3 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = t, & \\y & = 1 - 2t;\ t\in \mathbb{R} \\ \end{aligned}\)