For three vectors $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$, if $\mathbf{r}=3 \mathbf{p}+4 \mathbf{q}$ and…

For three vectors $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$, if $\mathbf{r}=3 \mathbf{p}+4 \mathbf{q}$ and $2 \mathbf{r}=\mathbf{p}-3 \mathbf{q}$, then
  1. $|\mathbf{r}| < 2|\mathbf{q}|$ and $\mathbf{r}, \mathbf{q}$ have the same direction
  2. $|\mathbf{r}|>2|\mathbf{q}|$ and $\mathbf{r}, \mathbf{q}$ have opposite directions
  3. $|\mathbf{r}| < 2|\mathbf{q}|$ and $\mathbf{r}, \mathbf{q}$ have opposite directions
  4. $|\mathbf{r}|>2|\mathbf{q}|$ and $\mathbf{r}, \mathbf{q}$ have the same direction

Solution

For three vectors $p, q$ and $r$
From eqs. (i) and (ii), we get $\begin{aligned} & r=3(2 r+3 q)+4 q \\ & r=6 r+9 q+4 q \\ & \Rightarrow \quad 5 r=-13 q \Rightarrow r=\frac{13}{5} q \Rightarrow|r|=\frac{13}{5}|q| \end{aligned}$ $\therefore \quad r$ and $q$ have opposite directions Hence, $|r|>2|q|$

Asked in: AP EAMCET 2016

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