How to approach solving for parallel vectors?

[pavadrin]

Hi
I’m given a series of problems in which I need to find the vector equation of a line which pass through a given point and are parallel to another vector.
The first problem:
Find a parallel vector to b = 5i-j, and passes through the point with position vector a = 2i+3j
My working:
r = a + λb
= (2i+3j) + λ(5i-j)
= (2+5λ)i + (3-λ)j
From this point onwards I do not have the slightest clue on where to go next, or if what I have done is correct. Could somebody please tell me if what I am doing is correct or incorrect, and if so, the correct way to approach these pesky problems.
Thanks
Pavadrin

 

[danago]

Looks fine to me so far, unless you need to write it as parametic or cartesian equations

From this point onwards I do not have the slightest clue on where to go next, or if what I have done is correct. Could somebody please tell me if what I am doing is correct or incorrect, and if so, the correct way to approach these pesky problems.
Thanks

Well that depends on what the question asks If it just asks for the vector equation of the line, then that's it.

Hi Nikki :D

 

[Architeuthis Dux]

Hi Pavadrin,

Solving for parallel vectors can be tricky, but your approach is definitely on the right track. The vector equation you have set up is correct, and the next step would be to plug in the given point a = 2i+3j to solve for the value of λ.

r = a + λb
= (2i+3j) + λ(5i-j)
= (2+5λ)i + (3-λ)j

To solve for λ, we can equate the x and y components of the vector equation to the x and y components of the given point. This means setting up the following equations:

2+5λ = x (x-component of the given point)
3-λ = y (y-component of the given point)

From here, we can solve for λ by rearranging the equations:

5λ = x - 2
λ = (x-2)/5

-λ = y - 3
λ = 3-y

Since both equations have the same value for λ, we can set them equal to each other and solve for x and y:

(x-2)/5 = 3-y
x = 15-5y

Now, we can plug in these values for x and y into the vector equation to get the final equation for the parallel vector:

r = (15-5y)i + (3-y)j

I hope this helps you understand the process better. Keep practicing and you'll become a pro at solving for parallel vectors in no time! Let me know if you have any other questions. Good luck!