Question

A sum of scalar multiples of two or more vectors​ (such as c 1 Bold u plus c 2 Bold v plus c 3 Bold wc1u+c2v+c3w​, where c Subscript ici are​ scalars) is called a linear combination of the vectors. Let iequals=left angle 1 comma 0 right angle1,0 and jequals=left angle 0 comma 1 right angle0,1. Express left angle negative 4 comma 9 right angle−4,9 as a linear combination of i and j ​(that is, find scalars c 1c1 and c 2c2 such that left angle negative 4 comma 9 right angle−4,9equals=c 1c1iplus+c 2c2j​).

Answer 1

-4<1,0>+9<0,1>=<-4,9>

Explanation:

i=<1,0>

j=<0,1>

We are required to find scalar multiples c₁ and c₂ such that

c₁i+c₂j=<-4,9>

c₁<1,0>+c₂<0,1>=<-4,9>

<c₁,0>+<0,c₂>=<-4,9>

This implies that

c₁+0=-4

c₁=-4

0+c₂=9

c₂=9

Since we have found the values of the scalar multiples c₁ and c₂, we next express <-4,9> as a linear combination of i and j.

This is given as:

-4<1,0>+9<0,1>=<-4,9>