# Thread: Mechanics, solving λ throught j and i

1. ## Mechanics, solving λ throught j and i

Given that a=i-2j and b=-3i+j find λ if a+λb is parallel to -i-3j.
please help. I have my a-level exam in 2 days and this question is killing me, plus the mechanics book doesnt even show how solve these questions.

2. Originally Posted by keaton
Given that a=i-2j and b=-3i+j find λ if a+λb is parallel to -i-3j.
First, write down a+λb, namely (i–2j) + λ(–3i+j) = (1–3λ)i + (–2+λ)j. Then use the fact that if two vectors are parallel then they are scalar multiples of each other. So we must have (1–3λ)i + (–2+λ)j = μ(–i–3j) for some value of μ. Now compare coefficients of i and j on both sides, to get 1–3λ = –μ and –2+λ = –3μ. Those are simple simultaneous equations for λ and μ, which you can solve for λ.

3. Substitute th valus of a and b in a + λb.
Collect the terms containing i and j. Take the ratio of their coefficints with that of -i - 3j. and equate them to a constant k. From that find λ. .

4. Originally Posted by keaton
Given that a=i-2j and b=-3i+j find λ if a+λb is parallel to -i-3j.
please help. I have my a-level exam in 2 days and this question is killing me, plus the mechanics book doesnt even show how solve these questions.

a+λb = (1-3λ)i + (λ-2)j = n(-i-3j) {ie any vector parallel to -i-3j}

Compare coeffiecients:
i's : -n = (1-3λ) (#)
j's : -3n = λ-2 (*)

So from (#) n = 3λ-1, substitute into (*) to get λ

5. Thanks for the help.

i used a+λb
then written in terms of i and j gives:
(1-3λ)i + (λ-2)j=-i-3j

so
(i)=-1
(j)=-3

to make them the same *(i) by 3 giving: (and if a is parallel to be a=b)
3(1-3λ)=(λ-2)
which symplifies to get:
3-9λ=λ-2
so λ=1/2

6. Originally Posted by keaton
Thanks for the help.

i used a+λb
then written in terms of i and j gives:
(1-3λ)i + (λ-2)j=-i-3j No!. Those vectors are not equal, they are parallel, which means that they are scalar multiples of each other. Look at the previous comments to see how to deal with that.

so