Question

If $\vec{a} = 2\hat{i} - \hat{j} - m\hat{k}$ and $\vec{b} = \frac{4}{7}\hat{i} - \frac{2}{7}\hat{j} + 2\hat{k}$ are collinear, then the value of $m$ is equal to:

• A. $-7$
• B. $-1$
• C. $2$
• D. $7$
• E. $-2$

Hint:

Collinear vectors are just parallel vectors. If you can visually see that multiplying one vector by a constant gives the other, you can avoid the full ratio calculation. Here, $\vec{a} = 3.5 \times \vec{b}$.

Answer 1

The Correct Option is A

Concept: Two vectors $\vec{a}$ and $\vec{b}$ are collinear if one is a scalar multiple of the other ($\vec{a} = \lambda\vec{b}$). This implies that the ratios of their corresponding components must be equal.

Step 1: Set up the ratio of the components.
Given the vectors from the image: $\vec{a} = (2, -1, -m)$ and $\vec{b} = \left(\frac{4}{7}, -\frac{2}{7}, 2\right)$. The condition for collinearity is: \[ \frac{2}{4/7} = \frac{-1}{-2/7} = \frac{-m}{2} \]

Step 2: Calculate the common ratio $\lambda$.
Using the first two components: \[ \lambda = \frac{2 \times 7}{4} = \frac{14}{4} = \frac{7}{2} \] Check with the second component: \[ \lambda = \frac{-1 \times 7}{-2} = \frac{7}{2} \] The ratio is consistent.

Step 3: Solve for $m$.
Equate the third component ratio to $\lambda$: \[ \frac{-m}{2} = \frac{7}{2} \] Multiplying both sides by $2$: \[ -m = 7 \quad \Rightarrow \quad m = -7 \]