Question

If the vectors $\vec{a} = 2\hat{i} + p\hat{j} + 4\hat{k}$ and $\vec{b} = 6\hat{i} - 9\hat{j} + q\hat{k}$ are collinear, then $p$ and $q$ are

• A. $p = 3, q = -12$
• B. $p = 3, q = 12$
• C. $p = -3, q = 12$
• D. $p = -3, q = -12$

Hint:

To solve collinear vector problems quickly by inspection, look at the scaling factor between the known numbers. Here, the $\hat{i}$ component increases from 2 to 6, which is exactly a $3\times$ scale factor ($\vec{b} = 3\vec{a}$). Therefore, simply multiply across to find the missing values: $-9 = 3p \implies p = -3$, and $q = 3(4) \implies q = 12$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem states that two vectors, $\vec{a}$ and $\vec{b}$, are collinear (parallel). We need to determine the values of the unknown scalar components $p$ and $q$.

Step 2: Key Formula or Approach:
If two vectors $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are collinear, their corresponding directional coefficients must be directly proportional: $$\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}$$

Step 3: Detailed Explanation:
Extract the coefficients from the given vectors:

• For $\vec{a}$: $a_1 = 2, a_2 = p, a_3 = 4$

• For $\vec{b}$: $b_1 = 6, b_2 = -9, b_3 = q$
Set up the proportionality ratios: $$\frac{2}{6} = \frac{p}{-9} = \frac{4}{q}$$ Simplify the known reference ratio: $$\frac{2}{6} = \frac{1}{3}$$ Now, solve for $p$ and $q$ separately by equating them to $\frac{1}{3}$:

• Solving for $p$: $$\frac{1}{3} = \frac{p}{-9} \implies p = \frac{-9}{3} \implies p = -3$$

• Solving for $q$: $$\frac{1}{3} = \frac{4}{q} \implies q = 4 \times 3 \implies q = 12$$
This gives the values $p = -3$ and $q = 12$, which matches option (C).

Step 4: Final Answer:
The component values are $p = -3$ and $q = 12$, corresponding to option (C).