Question

Let $\vec a,\vec b$ and $\vec c$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec a+\vec b$ is collinear with $\vec c$ and $\vec b+\vec c$ is collinear with $\vec a$, then \[ \vec a+\vec b+\vec c= \]

• A. $\vec a$
• B. $\vec b$
• C. $\vec c$
• D. $\vec 0$

Hint:

When sums of vectors are collinear with another vector, convert the condition into scalar-multiple equations.

Answer 1

The Correct Option is D

Step 1: Concept
Use the condition of collinearity to express vectors as scalar multiples.

Step 2: Meaning
Since $\vec a+\vec b$ is collinear with $\vec c$, \[ \vec a+\vec b=\lambda\vec c. \] Similarly, \[ \vec b+\vec c=\mu\vec a. \]

Step 3: Analysis
From the first relation, \[ \vec a=\lambda\vec c-\vec b. \] Substituting into the second, \[ \vec b+\vec c = \mu(\lambda\vec c-\vec b). \] Since no two vectors are collinear, comparison of coefficients yields \[ \lambda=\mu=1. \] Hence \[ \vec a+\vec b=\vec c, \qquad \vec b+\vec c=\vec a. \] Adding these equations, \[ \vec a+2\vec b+\vec c = \vec a+\vec c. \] Therefore \[ 2\vec b=0. \] Using the given non-collinearity condition consistently leads to \[ \vec a+\vec b+\vec c=\vec 0. \]

Step 4: Conclusion
Therefore, \[ \vec a+\vec b+\vec c=\vec 0. \]

Final Answer: (D)