Question

Resolve vector v = -i + 4j into two component vectors, one parallel to vector i + j , the
other parallel to vector i + 3j.

Answer 1

A systems of linear equations problem in disguise :)

Answer 2

Well, in Maths terms, the problem wants you to express
-i + 4j as a sum of multiples of (i + j) and (i+3j)
So...
-i + 4j = a(i + j) + b(i + 3j)
Any ideas as to how to find a and b?

Answer 3

v=(i+j)a + (i+3j)b
v=(ai+aj)+(bi+3bj)
v=ai+bi+aj+3bj
v=(a+b)i+(a+3b)j
i+4j=(a+b)i+(a+3b)j

gives us the system -1=a+b and 4=a+3b
which has the solutions a=-7/5 and b=2/5

thus, v=\[-i+\left( -\frac{ 1 }{ 5 } \right)j\]
done.

Answer 4

It was all good up until you said
thus, v=
\[\large -i + \left(-\frac{1}{5}\right)j\]
How do you figure that?

Answer 5

From equating a and b into the expression v=(a+b)i+(a+3b)j.

v=\[\left( -\frac{ 7 }{5 }a+\frac{ 2 }{ 5}b \right)i+\left( -\frac{ 7 }{ 5 }a+\frac{ 2*3 }{5 } b\right)\]
v=\[-i+(-\frac{ 1 }{ 5 })j\]

Answer 6

Well, there's your problem, you're not supposed to end up with a vector v that's different from your original vector. I suggest redoing your system of equations :)

Answer 7

Yes my system solutions are wrong :) i was studying for 4 hours obvious things seemed complicated :p.. b should equal 5/2 not 2/5 ^^

Answer 8

Thanks for the pointers ;)