Question

the two unit vectors parallel to the line 𝑦=−(3𝑥+5)

Answer 1

We have for coordinates of the vector $\vec u:$ $y=-3x.$ Then

$$|\vec u|=\sqrt{x^2+y^2}=\sqrt{x^2+(-3x)^2}=|x|\sqrt{10}=1$$

$$|x|=1/\sqrt{10}$$

The two unit vectors parallel to the line $𝑦=−(3𝑥+5)$ are

$$\vec u_1=\lang -1/\sqrt{10}, 3/\sqrt{10}\rangle, \vec u_2=\lang 1/\sqrt{10}, -3/\sqrt{10}\rangle$$