Question

Let the three vectors $\vec{a}, \vec{b}$, and $\vec{c}$ be pairwise non-collinear vectors. If $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$ and if $\vec{b}+2\vec{c}$ is collinear with $\vec{a}$, then $\vec{a}+2\vec{b}+5\vec{c}$ is equal to

• A. $\vec{0}$
• B. $\vec{a}$
• C. $\vec{b}$
• D. $\vec{c}$
• E. $2\vec{a}$

Hint:

Logic Tip: A common property in such vector collinearity problems is that if $\vec{x} + \alpha\vec{y}$ is parallel to $\vec{z}$ and $\vec{y} + \beta\vec{z}$ is parallel to $\vec{x}$, their combined linear sum $(\vec{x} + \alpha\vec{y} + \alpha\beta\vec{z})$ often equals the zero vector $\vec{0}$. This instantly shows $\vec{a} + 2\vec{b} + 4\vec{c} = \vec{0}$, making $\vec{a} + 2\vec{b} + 5\vec{c} = \vec{c}$.

Answer 1

The Correct Option is D