Let \(\mathbf{u}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}\) and \(\mathbf{v}=-3 \hat{\mathbf{i}}+5…
Let \(\mathbf{u}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}\) and \(\mathbf{v}=-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}\). Consider three points \(P, Q\) and \(R\) having position vectors \(\frac{-1}{7} \hat{\mathbf{i}}, \frac{-1}{4} \hat{\mathbf{j}}\) and \(-2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) respectively.
Among these, the points in the line segment passing through \(\mathbf{u}\) and \(\mathbf{v}\) are
Only \(P\) and \(Q\)
Only \(P\) and \(R\)
Only \(Q\) and \(R\)
All \(P, Q\) and \(R\)
Solution
Given vectors \(\mathbf{u}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}\) and \(\mathbf{v}=-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}\).
Now, equation of line segment passes through the points \(\mathbf{u}\) and \(\mathbf{v}\) is
\(\frac{x-1}{4}=\frac{y+2}{-7}\) ...(i)
Now, the point vectors having position vectors \(\mathbf{O P}=-\frac{1}{7} \hat{\mathbf{i}}, \mathbf{O Q}=-\frac{1}{4} \hat{\mathbf{j}}\) and \(\mathbf{O R}=-2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) are given.
The point \(P\left(-\frac{1}{7}, 0\right)\) by putting in line (i)
\(\frac{-\frac{1}{7}-1}{4}=\frac{2}{-7} \Rightarrow \frac{-8}{4 \times 7}=\frac{2}{-7} \Rightarrow \frac{-2}{7}=\frac{2}{-7}\)
\(\because\) Point \(P\left(-\frac{1}{7}, 0\right)\) satisfied the line, so it is on the line segment (i)
The point \(Q\left(0,-\frac{1}{4}\right)\) by putting in line (i)
\(\frac{0-1}{4}=\frac{-\frac{1}{4}+2}{7} \Rightarrow-\frac{1}{4}=-\frac{1}{4}\)
\(\because\) Point \(Q\left(0,-\frac{1}{4}\right)\) also satisfied the line (i), so it lies on the line segment.
Now, the point \(R(-2,3)\), on putting in the equation of line (i) \(\frac{-2-1}{4} \neq \frac{3+2}{-7}\)
\(\because\) Point \(Q(-2,3)\) not satisfied the line (i), so it doesn't line on the line segment.
Hence, option (a) is correct.