Question:
Show that line \( AB \) passing through points \( A(0, 4, 1), B(2, 3, -1) \) and the line \( CD \) passing through points \( C(4, 5, 0), D(2, 6, 2) \) are parallel. Also, find distance between them.
Show that line \( AB \) passing through points \( A(0, 4, 1), B(2, 3, -1) \) and the line \( CD \) passing through points \( C(4, 5, 0), D(2, 6, 2) \) are parallel. Also, find distance between them.
Show Hint
Parallel lines have direction ratios \( a, b, c \) and \( ka, kb, kc \).
Distance can also be found using the projection method: \( \sqrt{|\vec{CP}|^2 - (\text{proj}_{\vec{b}}\vec{CP})^2} \).
Distance can also be found using the projection method: \( \sqrt{|\vec{CP}|^2 - (\text{proj}_{\vec{b}}\vec{CP})^2} \).
Updated On: Sep 10, 2026
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Solution and Explanation
Concept:
• Direction Ratios (DRs): DRs of line through \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) are \( (x_2-x_1, y_2-y_1, z_2-z_1) \).
• Parallel Lines: Two lines are parallel if their DRs are proportional.
• Distance between parallel lines: \( d = \frac{|\vec{CP} \times \vec{d}|}{|\vec{d}|} \), where \( P \) is on line \( AB \), \( C \) is on line \( CD \), and \( \vec{d} \) is the direction vector.
Step 1: Determine the direction ratios of both lines
For line \( AB \): \( DRs = (2 - 0, 3 - 4, -1 - 1) = (2, -1, -2) \).
For line \( CD \): \( DRs = (2 - 4, 6 - 5, 2 - 0) = (-2, 1, 2) \). Comparing the DRs: \[ \frac{2}{-2} = \frac{-1}{1} = \frac{-2}{2} = -1 \] Since the DRs are proportional, the lines \( AB \) and \( CD \) are parallel.
Step 2: Set up vector quantities for distance calculation
Let point \( P = A(0, 4, 1) \) on line \( AB \).
Let point \( C = (4, 5, 0) \) on line \( CD \).
Vector \( \vec{CP} = (0 - 4)\hat{i} + (4 - 5)\hat{j} + (1 - 0)\hat{k} = -4\hat{i} - \hat{j} + \hat{k} \).
Direction vector of line \( CD \), \( \vec{b} = -2\hat{i} + \hat{j} + 2\hat{k} \).
Step 3: Calculate the distance
Cross product \( \vec{CP} \times \vec{b} \): \[ \vec{CP} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -4 & -1 & 1 \\ -2 & 1 & 2 \end{vmatrix} = \hat{i}(-2 - 1) - \hat{j}(-8 + 2) + \hat{k}(-4 - 2) = -3\hat{i} + 6\hat{j} - 6\hat{k} \] Magnitude \( |\vec{CP} \times \vec{b}| = \sqrt{(-3)^2 + 6^2 + (-6)^2} = \sqrt{9 + 36 + 36} = \sqrt{81} = 9 \).
Magnitude \( |\vec{b}| = \sqrt{(-2)^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = 3 \). \[ \text{Distance} = \frac{|\vec{CP} \times \vec{b}|}{|\vec{b}|} = \frac{9}{3} = 3 \text{ units} \]
• Direction Ratios (DRs): DRs of line through \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) are \( (x_2-x_1, y_2-y_1, z_2-z_1) \).
• Parallel Lines: Two lines are parallel if their DRs are proportional.
• Distance between parallel lines: \( d = \frac{|\vec{CP} \times \vec{d}|}{|\vec{d}|} \), where \( P \) is on line \( AB \), \( C \) is on line \( CD \), and \( \vec{d} \) is the direction vector.
Step 1: Determine the direction ratios of both lines
For line \( AB \): \( DRs = (2 - 0, 3 - 4, -1 - 1) = (2, -1, -2) \).
For line \( CD \): \( DRs = (2 - 4, 6 - 5, 2 - 0) = (-2, 1, 2) \). Comparing the DRs: \[ \frac{2}{-2} = \frac{-1}{1} = \frac{-2}{2} = -1 \] Since the DRs are proportional, the lines \( AB \) and \( CD \) are parallel.
Step 2: Set up vector quantities for distance calculation
Let point \( P = A(0, 4, 1) \) on line \( AB \).
Let point \( C = (4, 5, 0) \) on line \( CD \).
Vector \( \vec{CP} = (0 - 4)\hat{i} + (4 - 5)\hat{j} + (1 - 0)\hat{k} = -4\hat{i} - \hat{j} + \hat{k} \).
Direction vector of line \( CD \), \( \vec{b} = -2\hat{i} + \hat{j} + 2\hat{k} \).
Step 3: Calculate the distance
Cross product \( \vec{CP} \times \vec{b} \): \[ \vec{CP} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -4 & -1 & 1 \\ -2 & 1 & 2 \end{vmatrix} = \hat{i}(-2 - 1) - \hat{j}(-8 + 2) + \hat{k}(-4 - 2) = -3\hat{i} + 6\hat{j} - 6\hat{k} \] Magnitude \( |\vec{CP} \times \vec{b}| = \sqrt{(-3)^2 + 6^2 + (-6)^2} = \sqrt{9 + 36 + 36} = \sqrt{81} = 9 \).
Magnitude \( |\vec{b}| = \sqrt{(-2)^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = 3 \). \[ \text{Distance} = \frac{|\vec{CP} \times \vec{b}|}{|\vec{b}|} = \frac{9}{3} = 3 \text{ units} \]
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