*IBV5 The vectors u, v are given by u = 3i + 5j, v = i – 2j

[karush]

The vectors $u, v$ are given by $u = 3i + 5j, v = i – 2j$
Find scalars $a, b$ such that $a(u + v) = 8i + (b – 2)j$

$(u+v)=4i+3j$
in order to get the $8i$ let $a=2$
then $2(4i+3j)=8i+6j$
in order to get $6j$ let $b=8$ then $(8-2)j=6j$
so $a=2$ and $b=8$

I am not sure of the precise definition of what scalar means here...at least with vectors

 

[tkhunny]

vectors u,v are given by u=3i+5j,v=i–2j
Find scalars a,b such that a(u+v)=8i+(b–2)j

u + v = <4,3>

Then <4a,3a> = <8,(b-2)>

4a = 8

3a = b-2

 

[eddybob123]

I am not sure of the precise definition of what scalar means here...at least with vectors

The definition is the same with everything; basically a number. Whether it's restricted to real or complex numbers is usually clear from context.

 

[karush]

vectors u,v are given by u=3i+5j,v=i–2j
Find scalars a,b such that a(u+v)=8i+(b–2)j

u + v = <4,3>

Then <4a,3a> = <8,(b-2)>

4a = 8

3a = b-2

yes, that looks like a much better way to solve it. especially if it gets a lot more complicated.

 

[CellCoree]

, scalars are simply the coefficients that multiply the unit vectors. So, in this case, a scalar would be any real number that can be multiplied to a vector to produce another vector. In this problem, we are given the vectors $u$ and $v$ and we need to find scalars $a$ and $b$ such that when we multiply $a(u+v)$, we get the vector $8i+(b-2)j$.

To find these scalars, we can first simplify the expression $a(u+v)$ by distributing the scalar $a$ to each term in the parentheses. This gives us $au+av$. Then, we can plug in the given values for $u$ and $v$ to get $a(3i+5j)+a(i-2j)$.

To get the desired vector $8i+(b-2)j$, we need the $i$ component to be $8i$. This means we need to have $a(3i)+a(i)=8i$. This can be achieved if $a=2$. Similarly, for the $j$ component to be $(b-2)j$, we need to have $a(5j)+a(-2j)=(b-2)j$. This can be achieved if $b=8$.

Therefore, the scalars $a=2$ and $b=8$ satisfy the given condition and the final expression becomes $2(u+v)=8i+6j$, which matches the desired vector.