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: Understanding the Question:
We are given two vector linear combinations and need to determine the directional relationships between vectors $\vec{a}, \vec{b},$ and $\vec{c}$.
Step 3: Detailed Explanation:
1. Substitute $\vec{c}$ from the first equation into the second:
$3(5\vec{a} + 6\vec{b}) = \vec{a} - 4\vec{b}$
$15\vec{a} + 18\vec{b} = \vec{a} - 4\vec{b}$
$14\vec{a} = -22\vec{b} \implies \vec{a} = -\frac{11}{7} \vec{b}$.
Since the scalar is negative, $\vec{a}$ and $\vec{b}$ are in opposite directions.
2. Now express $\vec{c}$ in terms of $\vec{b}$:
$\vec{c} = 5(-\frac{11}{7}\vec{b}) + 6\vec{b} = (-\frac{55}{7} + \frac{42}{7})\vec{b} = -\frac{13}{7} \vec{b}$.
This shows $\vec{c}$ and $\vec{b}$ are also in opposite directions.
3. Express $\vec{a}$ in terms of $\vec{c}$:
$\vec{a} = -\frac{11}{7} \vec{b}$ and $\vec{b} = -\frac{7}{13} \vec{c}$
$\vec{a} = -\frac{11}{7} (-\frac{7}{13} \vec{c}) = \frac{11}{13} \vec{c}$.
Since the scalar is positive, $\vec{a}$ and $\vec{c}$ are in the same direction.
Step 4: Final Answer:
$\vec{a}$ and $\vec{c}$ are in the same direction, but $\vec{a}$ and $\vec{b}$ are in opposite directions.