Question

If $$a$$ and $$b$$ are two non collinear vectors and $$x,y$$ are two scalars such that $$\overrightarrow{a}x+\overrightarrow{b}y=0$$ this implies that:
A. $$x=y=-1$$
B. $$x=y=0$$
C. $$x=y=1$$
D. $$x=y=i$$

Answer 1

Hint: In this question, we will go for option verification and find out for which values of $$x,y$$ the given two vectors $$a$$ and $$b$$are non-collinear. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Given $$a$$ and $$b$$ are two non-collinear vectors and $$x,y$$ are two scalars.
Also, $$\overrightarrow{a}x+\overrightarrow{b}y=0$$
That implies $$\overrightarrow{a}=-{\displaystyle \frac{y}{x}}\overrightarrow{b}..................................\left(1\right)$$
If $$x$$ and $$y$$ are non-zero, then the two vectors $$a$$ and $$b$$ are collinear, because $$\overrightarrow{a}=\lambda \overrightarrow{b}$$ where $$\lambda$$ is scalar as shown in the below figure:

Now, we will go for the option verification to check whether the two vectors $$a$$ and $$b$$ are collinear or not.
A. By substituting $$x=y=-1$$ in equation (1), we have
$$\begin{gathered} \Rightarrow \overrightarrow{a}=-{\displaystyle \frac{-1}{-1}}\overrightarrow{b} \\ \therefore \overrightarrow{a}=-\overrightarrow{b} \end{gathered}$$
which is of the form $$\overrightarrow{a}=\lambda \overrightarrow{b}$$. So, for the values of $$x=y=-1$$, $$a$$ and $$b$$ are collinear vectors.
B. By substituting $$x=y=0$$ in equation (1), we have
$$\Rightarrow \overrightarrow{a}=-{\displaystyle \frac{0}{0}}\overrightarrow{b}$$
which is an indeterminate form. So, for the values of $$x=y=0$$, $$a$$ and $$b$$ are non-collinear vectors.
C. By substituting $$x=y=1$$ in equation (1), we have
$$\begin{gathered} \Rightarrow \overrightarrow{a}=-{\displaystyle \frac{1}{1}}\overrightarrow{b} \\ \therefore \overrightarrow{a}=-\overrightarrow{b} \end{gathered}$$
which is of the form $$\overrightarrow{a}=\lambda \overrightarrow{b}$$. So, for the values of $$x=y=1$$, $$a$$ and $$b$$ are collinear vectors.
D. By substituting $$x=y=i$$ in equation (1), we have
$$\begin{gathered} \Rightarrow \overrightarrow{a}=-{\displaystyle \frac{i}{i}}\overrightarrow{b} \\ \therefore \overrightarrow{a}=-\overrightarrow{b} \end{gathered}$$
which is of the form $$\overrightarrow{a}=\lambda \overrightarrow{b}$$. So, for the values of $$x=y=i$$, $$a$$ and $$b$$ are collinear vectors.
Therefore, only for the values of $$x=y=0$$, the two vectors $$a$$ and $$b$$ are collinear.
Thus, the correct option is B. $$x=y=0$$

So, the correct answer is “Option B”.

Note: We can use any of the given conditions to prove the collinearity for two vectors:
1. Two vectors $$a$$ and $$b$$ are collinear if there exists a number $$n$$ such that $$\overrightarrow{a}=n\overrightarrow{b}$$.
2. Two vectors are collinear if relations of their coordinates are equal.
3. Two vectors are collinear if their cross product is equal to the zero vector.