Answer to Question #109923 in Analytic Geometry for Franco Joco

Question

Answer the two questions and show complete solution.

2. Find two unit vectors that are parallel to the

Answer 1

Let the line be y=3x+4

Let's find two unit vectors that are parallel to that line.

The slope the given line is 3.

Suppose that, this line makes an angle of $\theta$ with the +ve direction of the X−Axis.

Then, the unit vector $\overline{u}=(\cos\theta,\sin\theta)$ is parallel to the line.

By the definition of slope, we have

$\tan\theta=3$

therefore

$\sec^2\theta=1+\tan^2\theta=10$

$cos\theta=\frac{1}{\sec\theta}=\pm\frac{1}{\sqrt{10}}$

$\sin\theta=\sqrt{1-cos^2\theta}=\sqrt{1-\frac{1}{10}}=\pm\frac{3}{\sqrt{10}}$

Hence $\overline{u}(\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}})$ and $\overline{v}(-\frac{1}{\sqrt{10}},-\frac{3}{\sqrt{10}})$ are two vectors, parallel to the line y=3x+4