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Questions  >>  CBSE XII  >>  Math  >>  Vector Algebra
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Q)

The values of k for which $\mid k\overrightarrow{a}\mid\le\mid\overrightarrow{a}\mid$ and $k\overrightarrow{a}+\frac{1}{2}\overrightarrow{a}$ is parallel to $\overrightarrow{a}$ holds true are_________.

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A)
Toolbox:
  • Two vectors are parallel indicates that their magnitude cannot be zero.
  • $|k\overrightarrow a|=k |\overrightarrow a|$
It is given $|k\overrightarrow a|\leq\:|\overrightarrow a|$
=>$|k||\overrightarrow a|\leq\:|\overrightarrow a|$ or $k\leq\:1$
(ie)$-1\leq\:k\leq1$ Hence $k \in [-1,1]$
It is also given that $k\overrightarrow a+\large\frac{1}{2}$$\overrightarrow a$ is parallel to $\overrightarrow a$
This implies $k(\overrightarrow a+\large\frac{1}{2}$$ \overrightarrow a)=\large\frac{3k}{2}$$\overrightarrow a$ Hence $ k \neq 1/2$
Since $k\overrightarrow a$ is along the direction of $\overrightarrow a$ and its direction is not opposite to $\overrightarrow a$
The values of k for which $\mid k\overrightarrow{a}\mid\le\mid\overrightarrow{a}\mid$ and $k\overrightarrow{a}+\frac{1}{2}\overrightarrow{a}$ is parallel to $\overrightarrow{a}$ holds true are $k \in [-1,1] $ and $k \neq \large\frac{1}{2}$
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