Vectors p and q

**u2_wa** · March 22nd 2010, 02:38 AM

There are two coplanar vectors $p$ & $q$ . If $ap+bq=0$ (scalar multiplication), what can be inferred about the values of a & b?

In my view answer should be $a=b=0$ but how about if $p$ and $q$ are also parallel!

**Grandad** · March 22nd 2010, 04:55 AM

Hello u2_wa

Originally Posted by u2_wa

There are two coplanar vectors $p$ & $q$ . If $ap+bq=0$ (scalar multiplication), what can be inferred about the values of a & b?

In my view answer should be $a=b=0$ but how about if $p$ and $q$ are also parallel!

You are right: if $\vec p$ and $\vec q$ are non-zero, non-parallel vectors then $a = b = 0$ .

If $\vec p$ and $\vec q$ are parallel, then, for some $\lambda$ :

$\vec p =\lambda \vec q$

i.e. $\vec p -\lambda\vec q = \vec 0$ (note: a zero vector)

In which case, all you can say is that, for this value of $\lambda, \;a:b = -1:\lambda$ .

If $\vec p= \vec 0$ , and $\vec q \ne \vec 0$ , then $a\vec p + b\vec q = \vec 0 \Rightarrow b = 0$ and $a$ can take any value.

Similarly if $\vec p\ne \vec 0$ , and $\vec q = \vec 0$ , then $a\vec p + b\vec q = \vec 0 \Rightarrow a = 0$ and $b$ can take any value.

I think that just about covers all possible cases.

Grandad

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