Question

Given two force vectors:
\[ \vec{F}_1 = 2\hat{i} + 3\hat{j} - \hat{k}, \quad \vec{F}_2 = \hat{i} + \hat{j} + \hat{k} \] What is the magnitude of the resultant force?

• A. $ 3\ \text{N} $
• B. $ 4\ \text{N} $
• C. $ 5\ \text{N} $
• D. $ 6\ \text{N} $

Hint:

Always calculate the magnitude using the formula:
\[ |\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2} \] It’s essentially the 3D version of the Pythagorean theorem.

Answer 1

The Correct Option is C

Step 1: Add the vectors component-wise
To find the resultant force vector:
\[ \vec{F}_{\text{resultant}} = \vec{F}_1 + \vec{F}_2 = (2\hat{i} + 3\hat{j} - \hat{k}) + (\hat{i} + \hat{j} + \hat{k}) \] Add each component:
$\hat{i}: 2 + 1 = 3$
$\hat{j}: 3 + 1 = 4$
$\hat{k}: -1 + 1 = 0$
So,
\[ \vec{F}_{\text{resultant}} = 3\hat{i} + 4\hat{j} + 0\hat{k} \] Step 2: Find the magnitude of the resultant vector
The magnitude of a vector $\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$ is given by:
\[ |\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2} \] Here:
$F_x = 3$
$F_y = 4$
$F_z = 0$
So:
\[ |\vec{F}_{\text{resultant}}| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16} = \sqrt{25} = 5\ \text{N} \] Step 3: Match with the correct option
This matches option:
\[ (C) 5\ \text{N} \]