Parallel vector, I need a bit of explanation

[lioric]

TL;DR Summary

If we are making the vector parallel to i+3j, why do we multiply 3 with the i of the resultant vector ?

Given that c = 3i + 4j and d = i - 2j, find u if uc + d is parallel to i + 3j,

this is the question

in the solution,
it says that we have to multiply the 3 with the i

i do know that this is the ”method“ to do this question but I’d like a bit of explanatio.
I don’t understand why the 3 is multiplied with the i when in the question it says 3j.

thank you for your time.

 

[topsquark]

Hint:
What is the vector ## \mu c + d## in terms of i and j? What does it mean for one vector to be parallel to another?

-Dan

 

[DrClaude]

The answer the OP, vectors ai + bj and ci + dj will be parallel if b/a = d/c. So vector ai + bj will be parallel to i + 3j if b = 3a. This is where the factor 3 comes from.

 

[lioric]

Hint:
What is the vector ## \mu c + d## in terms of i and j? What does it mean for one vector to be parallel to another?

-Dan

If one vector is parallel that means the direction is the same. In terms of i and j the parallel vectors are always some multiple of the direction

 

[lioric]

The answer the OP, vectors ai + bj and ci + dj will be parallel if b/a = d/c. So vector ai + bj will be parallel to i + 3j if b = 3a. This is where the factor 3 comes from.

This speaks to me a lot. If something is parallel, the ratio of x to y lengths should be the same. Thank you very much. It makes so much more sense now.
I’ll propos this thread solved. Thank you all.
i love the way that you teach stuff