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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra1 Mark
MCQ
In triangle ABC (Fig 10.18), which of the following is not true:
  • A
    $\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=\vec{0}$
  • B
    $\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}-\overrightarrow{\text{AC}}=\vec{0}$
  • ✓
    $\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}-\overrightarrow{\text{CA}}=\vec{0}$
  • D
    $\overrightarrow{\text{AB}}-\overrightarrow{\text{CB}}+\overrightarrow{\text{CA}}=\vec{0}$
✓

Answer

Correct option: C.
$\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}-\overrightarrow{\text{CA}}=\vec{0}$

On applying the triangle law of addition in the given triangle, we have:
$\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}=\overrightarrow{\text{AC}}\ \ \ \ \ \ \ \ \ ....(1)$
$\Rightarrow\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}=-\overrightarrow{\text{CA}}$
$\Rightarrow\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=\vec{0}\ \ \ \ ....(2)$
$\therefore$ The equation given in alternative A is true.
$\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}=\overrightarrow{\text{AC}}$
$\Rightarrow\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}-\overrightarrow{\text{AC}}=\vec{0}$
$\therefore$ The equation given in alternative B is true.
From equation (2), we have:
$\overrightarrow{\text{AB}}-\overrightarrow{\text{CB}}+\overrightarrow{\text{CA}}=\vec{0}$
$\therefore$ The equation given in alternative D is true.
Now, consider the equation given in alternative C:
$\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}-\overrightarrow{\text{CA}}=\vec{0}$
$\Rightarrow\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}=\overrightarrow{\text{CA}}\ \ \ \ ....(3)$
From equations (1) and (3), we have:
$\overrightarrow{\text{AC}}=\overrightarrow{\text{CA}}$
$\Rightarrow\overrightarrow{\text{AC}}=-\overrightarrow{\text{AC}}$
$\Rightarrow\overrightarrow{\text{AC}}+\overrightarrow{\text{AC}}=\vec{0}$
$\Rightarrow2\overrightarrow{\text{AC}}=\vec{0}$
$\Rightarrow\overrightarrow{\text{AC}}=\vec{0},$ which is not true.
Hence, the equation given in alternative C is incorrect.
The correct answer is C.

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