Vectors 1 Question 23
24. The points with position vectors $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{a}}+k \overrightarrow{\mathbf{b}}$ are collinear for all real values of $k$.
(1984, 1M)
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Answer:
Correct Answer: 24. (b)
Solution:
- Let position vectors of points $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{C}}$ be $\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{a}}+k \overrightarrow{\mathbf{b}}$, respectively.
$$ \begin{array}{ll} \therefore & (\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}})-(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})=(\overrightarrow{\mathbf{a}}+k \overrightarrow{\mathbf{b}})-(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \\ \Rightarrow & -2 \overrightarrow{\mathbf{b}}=(k+1) \overrightarrow{\mathbf{b}} \\ \Rightarrow & k+1=-2 \\ \Rightarrow & k=-3 \end{array} $$
Hence, it is false statement.
