In a school, the swimming pool is shaped as a cuboid, and the breadth of the pool is represented by line \(\frac{-x+2}{-5} = y = z-3\). Two students A and B are swimming along the length of the pool.
Based on the above information, answer the following questions :
(i) If the path of student A is represented by line \(\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-3p}\), find the value of p.
(ii) Write the vector equation of the path of student B swimming parallel to the path of student A, if at a certain instant he passes through the point with coordinates \((2, 4, -1)\).
(iii) (a) Find the foot of the perpendicular drawn from \((2, 4, -1)\) to the path represented by student A.
Find the equations of the lines AC and BD, if coordinates of points A, B, C and D respectively, are \((3, -4, 2)\), \((-1, -2, 0)\), \((4, 1, -1)\) and \((4, 0, 5)\). Show that the lines AC and BD are perpendicular to each other.
Assertion (A) : Points \(A(6, -7, 0)\), \(B(16, -19, -4)\) and \(C(1, -1, 2)\) are collinear.
Reason (R) : Three points \(A\), \(B\) and \(C\) are collinear, when point \(C\) is the mid-point of line segment joining points \(A\) and \(B\).
Find the length of the perpendicular drawn from the point \( P(1, 2, 3) \) to the line \( \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2} \). Also, find the equation of the perpendicular line joining \( P \) and the foot of the perpendicular.
Find the co-ordinates of the point on the line \(\vec{r} = -\hat{j} + 3\hat{k} + \lambda(2\hat{i} - 2\hat{j} + \hat{k})\) such that the sum of co-ordinates is 3.
Prove that the line through points A(0, -1, -1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(-4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.
The length of perpendicular drawn from the point \((1, 2, 3)\) on line \(\dfrac{x}{0} = \dfrac{y}{1} = \dfrac{z}{0}\) is
Assertion (A) : Lines given by \(x = py + q\), \(z = ry + s\) and \(x = p^{\prime}y + q^{\prime}\), \(z = r^{\prime}y + s^{\prime}\) are perpendicular to each other when \(pp^{\prime} + rr^{\prime} = 1\).
Reason (R) : Two lines \(\vec{r} = \vec{a}_1 + \lambda\vec{b}_1\) and \(\vec{r} = \vec{a}_2 + \mu\vec{b}_2\) are perpendicular to each other if \(\vec{b}_1 \cdot \vec{b}_2 = 0\).
The length of perpendicular drawn from point \( (2, 5, 7) \) on line \( \dfrac{x}{1} = \dfrac{y}{0} = \dfrac{z}{0} \) is
Assertion (A) : Lines given by \( x = py + q \), \( z = ry + s \) and \( x = p^{\prime}y + q^{\prime} \), \( z = r^{\prime}y + s^{\prime} \) are perpendicular to each other when \( pp^{\prime} + rr^{\prime} = 1 \).
Reason (R) : Two lines \( \vec{r} = \vec{a}_1 + \lambda\vec{b}_1 \) and \( \vec{r} = \vec{a}_2 + \mu\vec{b}_2 \) are perpendicular to each other if \( \vec{b}_1 \cdot \vec{b}_2 = 0 \).