Q197 / 270 · NDA 2017 (Phase 2) · Mathematics

A force $\overrightarrow{\mathrm{F}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ acts on a particle to displace it from the point $\mathrm{A}(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ to the point $\mathrm{B}(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}})$. The work done by the force will be

  • A
    5 units
  • B
    7 units
  • Correct answer
    C
    9 units
  • D
    10 units

Solution

$\begin{aligned} & \overrightarrow{\mathrm{F}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{AB}}=(3-1) \hat{\mathrm{i}}+(-1-2) \hat{\mathrm{j}}+(5-(-3)) \hat{\mathrm{k}}\end{aligned}$ $=2 \hat{i}-3 \hat{j}+8 \hat{k}$ Work done $=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{AB}}=(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+8 \hat{\mathrm{k}})$ $\begin{aligned} & =(1 \times 2)+(3 \times(-3))+(2 \times 8) \\ & =2-9+16=9 \text { units. }\end{aligned}$

Hence, the correct answer is option: 'C'.

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