Question

If $\vec{c} = 5\vec{a} + 6\vec{b}$ and $3\vec{c} = \vec{a} - 4\vec{b}$ then}

• A. $\vec{a}, \vec{b}, \vec{c}$ are non-collinear
• B. $\vec{a}, \vec{b}, \vec{c}$ are in the same direction
• C. $\vec{a}, \vec{c}$ are in the same direction but $\vec{a}, \vec{b}$ are in the opposite direction
• D. $\vec{c}, \vec{b}$ are in the opposite direction and $\vec{a}, \vec{b}$ are in the same direction

Hint:

$\vec{u} = k\vec{v}$ means same direction if $k > 0$, opposite if $k < 0$.

Answer 1

The Correct Option is C

Step 1: Eliminate $\vec{c$}
Substitute $\vec{c}$ into second equation:
$3(5\vec{a} + 6\vec{b}) = \vec{a} - 4\vec{b}$
$15\vec{a} + 18\vec{b} = \vec{a} - 4\vec{b}$
$14\vec{a} = -22\vec{b} \implies \vec{a} = -\frac{11}{7}\vec{b}$.

Step 2: Analysis of $\vec{a$ and $\vec{b}$}
Since $\vec{a} = k\vec{b}$ with $k < 0$, $\vec{a}$ and $\vec{b}$ are in opposite directions.

Step 3: Relationship with $\vec{c$}
$\vec{c} = 5(-\frac{11}{7}\vec{b}) + 6\vec{b} = -\frac{55}{7}\vec{b} + \frac{42}{7}\vec{b} = -\frac{13}{7}\vec{b}$.
Since $\vec{c}$ is also a negative multiple of $\vec{b}$, it must be in the same direction as $\vec{a}$.
Final Answer: (C)