Linear dependence/independence

[masterofthewave124]

Given that and u||v, v is not parallel with w and z= −3u state (with reasons) whether the following vectors are linearly dependent with u.

a) -2v -> LD since u is parallel with v, u must be also parallel with 2v

b) 3w -> LI since u is parallel with v and v isn't parallel with w, w is also not parallel with u

c) 4z -> LD since u is parallel with v and v is parallel with z (z is just a scalar mutliple of u anyways), z must be parallel with u

d) 3v + 4z -> LD since v and z are parallel with each other and both are parallel with u, any combination of v and z is parallel with u

can someone check these for me? the coefficients infront of the vectors have no real siginificance right?, there just scalar multiples? if they do, then i screwed up big time.

 

[masterofthewave124]

different topic (division of a line) but i don't want to pollute the board with new threads.

1. State the ratio into which A divides BC. Indicate whether the division is internal or external.

a) BA = 2/3(BC)

is this question worded right? because you get BA/BC = 2/3 which really means B divides AC externally in the ratio 2:3. also, usually if C divides AB externally, the ratio would be AC:CB, which means one of them is negative. that's the second problem with this question even if the wording is incorrect.

 

[arunbg]

For the first question, I don't find anything wrong with your answers.
Remember that a scalar coefficient in front of a vector, simply scales the vector by some measure while maintaining its direction .

For the second question, if you are not talking of vectors, but merely scalars or line segments, both the answers are possible . However if it is vectors the situation changes and there is only one answer .
Before I explain, could you please clarify further .

 

[HallsofIvy]

I don't understand why you are worrying about "B divides AC externally" or "C divides AB externally". The problem asked in what ratio A divides BC. Clearly, A is 2/3 of the way from B to C and so divides BC 2 to 1. (BA is twice as long as AC)

 

[masterofthewave124]

HallsofIvy: yes thanks for clarifying, i needed to draw a diagram to visualize.

B______________A______C

it would look something like that i suppose.

 

[arunbg]

It can also be something like

A________B_______________C

Note that in this case too BA = 2/3 BC and division is external and in which case AB/BC = (2/3)/(1+2/3) = 2/5 .Thus both internal and external division are possible as I said in my earlier post.
Now had it been
$$\vec{BA} = \frac{2}{3}\vec{BC}$$
the situation would have been different .
Can you see how ? ;)