Find a vector equation and parametric equations of the line in $\mathbb{R}^2$ passing through the origin and is parallel to the vector $\vec{u}=(2,3)$

Question

anyone can help me? :<

Are there any equations that I could use in this question? I am so confused. I only know how to do the question if it changes "parallel" to "perpendicular" because I only know the equations for that...

thx

Answer 1

The vector equation for a line with direction $ \ \vec{u} \ = \ \langle a, b \rangle \ $ passing through a point $ \ (x_0, y_0) \ $ (written as a vector from the origin, $ \ \langle x_0 , y_0 \rangle \ ) , $ is

$$ \langle x, y \rangle \ = \ \langle x_0, y_0 \rangle \ + \ t \cdot \langle a,b \rangle \ \ . $$

Such an equation expresses the location of each point on the line as a vector extending from the origin to that point. (A line "parallel" to $ \ \vec{u} \ $ means that its direction is given by that vector.)

You will get the parametric form by writing out the corresponding equations for each component:

$$ x \ = \ x_0 \ + \ at \ \ , \ \ y = y_0 \ + \ bt \ . $$

Note that if you solve the first equation for $ \ t \ $ and insert that result into the second equation, you will obtain the standard "point-slope" equation for the line, $ \ ( y - y_0 ) \ = \ \frac{b}{a} \ (x - x_0) \ . $