Scalars and Vectors

Question

a = <2,3> and b = <1,-5>

Show that there are scalars S and T so that:

Sa + Tb = <-3,-24>

If you could explain how you solve each step, that would be extremely helpful.

Answer 1

To multiply a scalar times a vector simply multiply the scalar times each component of the vector.

To add two vectors add corresponding components.

So,

S<2,3> = <2S, 3S>

T<1,-5> = <T, -5T>

We want the result of adding those two vectors to be <-3,-24>.

2S + T = -3

3S - 5T = -24

There you have a system of two equations and two unknowns. Perhaps solving the first equation for T (T=-2S-3) and substituting into the 2nd equation for T is the way to go. You could also use addition/elimination.

3S -5(-2S-3) = -24

3S +10S + 15 = -24

13S = -39

S = -3

T = 3

Does -3<2,3> + 3<1,-5> = <-3, -24> ?

Answer 2

2S +T =-3

3S -5T = -24

6S +3T =-9

6S -10T =-48

subtract

13T =39

T =39/13 = 3

S = -3-3)/2 = -3

S=-3, T=3

<-6,-9> and <3,-15> sum to <-3,-24>