Question

If a and b are two non-collinear vectors and $$xa+yb=0$$ then
(A). $$x=0$$ but $$y$$ is not necessarily zero
(B). $$y=0$$ but $$x$$ is not necessarily zero
(C). $$x=0$$, $$y=0$$
(D). None of the above

Answer 1

Hint: Two vectors are said to be collinear if they lie on the same line or else they should be parallel so if two vectors are non-collinear then they should be anti- parallel that is the property of the vector we used in this problem. On the other hand non-collinear vectors are vectors lying in the same plane but they are not acting at the same point.

Complete step-by-step solution -
Given that a and b are two non-collinear vectors and $$xa+yb=0$$
$$\Rightarrow {\displaystyle \frac{a}{b}}={\displaystyle \frac{-y}{x}}$$. . . . . . . . . . . . . . . . . . . (1)
$$\Rightarrow a={\displaystyle \frac{-y}{x}}b$$
Two vectors a and b are said to be collinear if $$a=\lambda b$$
Given that a and b are two non-collinear vectors it is possible only when $$\lambda =0$$
Given that a and b are non collinear, so a and b should not be parallel, from equation (1) therefore $${\displaystyle \frac{y}{x}}=0$$ . . . . . . . . . . . . . . . . . . . (2)
From equation (2) we will get,
$$\Rightarrow y=0$$
The given equation can also be written as
$$\Rightarrow {\displaystyle \frac{b}{a}}={\displaystyle \frac{-x}{y}}$$ . . . . . . . . . . . . . . . . . . . (3)
$$\Rightarrow b={\displaystyle \frac{-x}{y}}a$$
Two vectors a and b are said to collinear if $$b=\lambda a$$
Given that a and b are two non-collinear vectors it is possible only when $$\lambda =0$$
Given that a and b are non-collinear, so a and b should not be parallel, from equation (3) therefore$${\displaystyle \frac{x}{y}}=0$$ . . . . . . . . . . . . . . . . . . . . . . . . (4)
From equation (4) we will get,
$$\Rightarrow x=0$$
Therefore the correct option for above question is option (C)

Note: Two vectors a and b are said to be collinear if $$b=\lambda a$$ or $$a=\lambda b$$ the value of $$\lambda$$ is any real number. If two vectors a and b are collinear then components of vector a and b are proportional and if not collinear they are not proportional.vector is a quantity which is described completely by magnitude as well as its direction. The physical quantities like displacement, velocity, acceleration, force are measured by their quantity as well as their direction so such types of quantities are vectors.