AP EAMCET · Maths · Three Dimensional Geometry
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
- A \(|\mathbf{r}| < 2|\mathbf{q}|\) and \(\mathbf{r}, \mathbf{q}\) have the same direction
- B \(|\mathbf{r}|>2|\mathbf{q}|\) and \(\mathbf{r}, \mathbf{q}\) have opposite directions
- C \(|\mathbf{r}| < 2|\mathbf{q}|\) and \(\mathbf{r}, \mathbf{q}\) have opposite directions
- D \(|\mathbf{r}|>2|\mathbf{q}|\) and \(\mathbf{r}, \mathbf{q}\) have the same direction
Answer & Solution
Correct Answer
(B) \(|\mathbf{r}|>2|\mathbf{q}|\) and \(\mathbf{r}, \mathbf{q}\) have opposite directions
Step-by-step Solution
Detailed explanation
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…
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