Check whether two vectors are parallel or perpendicular or none.

Question

Question :

Let ​a,b,c be three unit vectors such that 3a + 4b + 5c ​= 0.  Then which
of the following statements is true:
(a) a is parallel to b 
(b) ​a is perpendicular to b
(c) a is neither parallel nor perpendicular to b
(d) none of the above

From my understanding, since 3a + 4b + 5c ​= 0, therefore there are less than or equal to 2 independent vectors among a, b, c. Therefore a,b,c must be co-planer and may be parallel. Now, how to find whether it is parallel or not?

Answer 1

Hint:

If you see $3$,$4$, and $5$ in the same problem statement like this (or multiples of this like $6,8,10$ etc...) your mind should immediately be drawn to the example of a $3$-$4$-$5$ right-angled triangle. Try to think of how your example might be (and in fact must be) related to a $3$-$4$-$5$ triangle.

If $3a+4b+5c=0$ then we have that $3a+4b = -5c$. Taking the inner product of each side with itself, that is $\langle 3a + 4b, 3a+4b\rangle = \langle -5c,-5c\rangle$, we get...

$~$

$9\langle a,a\rangle + 24\langle a, b\rangle + 16\langle b,b\rangle = 25\langle c,c\rangle$ which simplifies further into... and implies that... which implies that...