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If \( \overrightarrow a\) and \( \overrightarrow b\) are two collinear vectors, then which of the following are incorrect:

\[ \begin{array}{l} (A) \quad \overrightarrow b = λ \overrightarrow a, \text{ for some scalar } λ \\(B) \quad \overrightarrow a = ± \overrightarrow b \\(C) \quad \text{the respective components of}\; \overrightarrow a\; and\; \overrightarrow b \text{ are not proportional} \\ (D) \quad \text{ both the vectors } \overrightarrow a\; and\; \overrightarrow b \text{ have same direction, but different magnitudes.} \end{array} \]

1 Answer

  • Two or more vectors ,if they are parallel or act along same direction are said to be collinear vectors.
  • Two vectors are collinear means they are parallel.
Step 1:
Ans A: Is correct.
Given that $\overrightarrow a$ and $\overrightarrow b$ are collinear.
Two vectors are said to be collinear if they act along same direction.
Hence $\overrightarrow b=\lambda\overrightarrow a$.
$\Rightarrow$ A is correct
Step 2:
Ans B: Is incorrect.
If $\overrightarrow a=\pm\overrightarrow b$ then their magnitudes should be equal.
But for collinear their magnitudes need not be equal.
Therefore, B need not be correct.
Step 3:
And C: Is correct
If two vectors are collinear then they are parallel.
If they are parallel then their corresponding components should be proportional
Therefore, C is correct.
Step 4:
Ans D: Is incorrect.
If two vectors have same direction then they are called like parallel vectors.
If they have opposite directions then they are called unlike parallel vectors.
Therefore, for parallel vectors their directions may be same or may be opposite.
Hence D is not necessary to be correct.
answered May 20, 2013 by sreemathi.v

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