Question

if a,b,c are three given coplanar vectors such that a=kb+jc .k and j are scalars then find k and j?

Answer 1

its hard!

Answer 2

so `a` is a linear combination of vectors `b` and `c`

Answer 3

if they are as shown, then k = j =1

Answer 4

If they are coplanar, then you an treat them like 2D vectors

Answer 5

But you only have two equations and like, eight variables, so you need more info.

Answer 6

essentially the question is asking you to solvea full column rank matrix when n=2 :

\[\begin{bmatrix} b_1& c_1\\b_2&c_2\\b_3&c_3\\\cdots\\\end{bmatrix}\begin{bmatrix}k\\j\end{bmatrix} = \begin{bmatrix} a_1\\a_2\\a_3\\\cdots \end{bmatrix}\]

Answer 7

assuming ofcourse b and c are heading in different directions
then you will have an unique solution

in 2D this solution equals [b c]^(-1) a

Answer 8

existence of unique solution is easy to see for a full column matrix as the right hand side is in columnspace of the left side matrix

Answer 9

@danish071996