Question

Two forces ${\mathrm{F}}_1=1\mathrm{N}$ and ${\mathrm{F}}_2=2\mathrm{N}$ act along the lines $\mathrm{x}=0$ and$\mathrm{y}=0$respectively. Then, the resultant of forces would be

• A. $\hat{\mathrm{i}}+2\hat{\mathrm{j}}$
• B. $\hat{\mathrm{i}}+\hat{\mathrm{j}}$
• C. $3\hat{\mathrm{i}}+2\hat{\mathrm{j}}$
• D. $2\hat{\mathrm{i}}+\hat{\mathrm{j}}$

Answer 1

The correct option is D

$2\hat{\mathrm{i}}+\hat{\mathrm{j}}$

The explanation for the correct option:

Option (d) $2\hat{\mathrm{i}}+\hat{\mathrm{j}}$

Step 1: Given data

First force: ${\mathrm{F}}_1=1\mathrm{N}$

Second force: ${\mathrm{F}}_2=2\mathrm{N}$

Given the line of action for the first force: $\mathrm{x}=0$

Given the line of action for the second force: $\mathrm{y}=0$

Step 2: Formula used

Vector addition:

1. Vector addition is the operation of adding two or more vectors together into a vector sum.
2. The so-called parallelogram law gives the rule for the vector addition of two or more vectors.

Parallelogram rule:

1. The parallelogram rule says that if we place two vectors so they have the same initial point, and then complete the vectors into a parallelogram, then the sum of the vectors is the directed diagonal that starts at the same point as the vectors.
2. So, $\mathrm{F}=\sqrt{{\mathrm{F}}_1^2+{\mathrm{F}}_2^2+2{\mathrm{F}}_1{\mathrm{F}}_2\mathrm{cos\theta }}$

Step 3: Find the Resultant force

First force: $\mathrm{x}=0$ means y-axis

${\overrightarrow{\mathrm{F}}}_1=\hat{\mathrm{ȷ}}$

Second force: $\mathrm{y}=0$ means x-axis

${\overrightarrow{\mathrm{F}}}_2=2\hat{\mathrm{j}}$

So, resultant according to the parallelogram law:

$\overrightarrow{\mathrm{F}}={\overrightarrow{\mathrm{F}}}_1+{\overrightarrow{\mathrm{F}}}_2=2\hat{\mathrm{i}}+\hat{\mathrm{j}}$

Hence, option (d) is the correct option.