If a parallelopiped is formed by planes drawn through the points $(5,8,10)$ and $(3,6,8)$ parallel to the coordinate planes, then the length of diagonal of the parallelopiped is (A) $2\sqrt{3}$ (B) $3\sqrt{2}$ (C) $\sqrt{2}$ (D) $\sqrt{3}$

If a parallelopiped is formed by planes drawn through the points (5,8,10) and (3,6,8) parallel to the coordina

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(5,8,10)

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Concept understanding — Collinearity Slope Condition

Collinearity Slope Condition — From Intuition to Precision

Imagine three points scattered on a sheet of paper. You can always draw a triangle through them. But if those three points happen to lie perfectly on a single straight line — like beads on a thread — they are called collinear (from Latin co- meaning "together" and linearis meaning "belonging to a line").

The question is: how do you check, using only coordinates, whether three given points are collinear?

The Intuition

Think about walking from point A to point B, then from point B to point C. If all three lie on the same line, your direction of travel should not change when you turn at B. In other words, the slope of AB must equal the slope of BC.

Slope measures steepness: $\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ . If two segments share the same slope and meet at a common point (B), they lie on the same straight line.

Note

This works because a line has a constant slope everywhere. If AB and BC have the same slope, they are parts of the same line — they cannot bend.

The Precise Statement

Let three points be $A(x_1, y_1)$ , $B(x_2, y_2)$ , and $C(x_3, y_3)$ . They are collinear if and only if:

\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2}

provided that $x_1 \neq x_2$ and $x_2 \neq x_3$ (i.e., no vertical segment).

\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2}

What About Vertical Lines?

If $x_1 = x_2$ , the slope of AB is undefined (division by zero). But the condition still works: if AB is vertical, then for collinearity, BC must also be vertical — meaning $x_2 = x_3$ . So the condition becomes: either both slopes are equal and defined, or both are undefined (i.e., both segments are vertical).

A Cleaner Algebraic Form

Cross-multiplying the slope equality gives a form that avoids division entirely:

(y_2 - y_1)(x_3 - x_2) = (y_3 - y_2)(x_2 - x_1)

This works for all cases, including vertical lines.

Tip

For quick checks, use the cross-multiplied form — no need to worry about zero denominators.

Example

Check if $A(1, 2)$ , $B(3, 6)$ , $C(5, 10)$ are collinear.

Slope of AB: $\frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$

Slope of BC: $\frac{10 - 6}{5 - 3} = \frac{4}{2} = 2$

Slopes are equal → points are collinear. Indeed, they all lie on $y = 2x$ .

Why This Matters

The slope condition is the simplest test for collinearity in coordinate geometry. It appears in: …

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A parallelopiped formed by planes parallel to coordinate planes is a rectangular cuboid. Given two opposite vertices, its side lengths are the absolute differences of their coordinates. The length of the diagonal is then found using the 3D distance formula, which results in $\boxed{2\sqrt{3}}$ .

When a parallelopiped is formed by planes drawn parallel to the coordinate planes, it means its faces are defined by equations like $x=k_1$ , $x=k_2$ , $y=k_3$ , $y=k_4$ , $z=k_5$ , $z=k_6$ . This specific type of parallelopiped is a rectangular cuboid (or a rectangular box), because its edges are parallel to the coordinate axes, making all its internal angles $90^\circ$ .

The two given points, $(5,8,10)$ and $(3,6,8)$ , represent opposite vertices of this rectangular cuboid. Imagine one point as the "bottom-front-left" corner and the other as the "top-back-right" corner. The difference in their respective coordinates will give us the dimensions (length, width, height) of the cuboid.

Identify the coordinates of the given points.

Let the two given points be $P_1 = (x_1, y_1, z_1) = (5, 8, 10)$ and $P_2 = (x_2, y_2, z_2) = (3, 6, 8)$ .
Determine the dimensions of the parallelopiped (cuboid).

The side lengths of the cuboid are the absolute differences between the corresponding coordinates of the two opposite vertices.
- Length ( $a$ ) along the x-axis: $a = |x_2 - x_1|$
- Width ( $b$ ) along the y-axis: $b = |y_2 - y_1|$
- Height ( $c$ ) along the z-axis: $c = |z_2 - z_1|$
Calculating these values:
- $a = |3 - 5| = |-2| = 2$ units
- $b = |6 - 8| = |-2| = 2$ units
- $c = |8 - 10| = |-2| = 2$ units
So, the parallelopiped is a cube with side length $2$ units.

Tip

The order of subtraction for finding the absolute difference does not matter, as $|k| = |-k|$ . For example, $|5-3| = 2$ and $|3-5| = 2$ .
Recall the formula for the length of the diagonal of a cuboid. …

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Q.If a parallelopiped is formed by planes drawn through the points (5,8,10)(5,8,10)(5,8,10) and (3,6,8)(3,6,8)(3,6,8) parallel to the coordinate planes, then the length of diagonal of the parallelopiped is (A) 232\sqrt{3}23​ (B) 323\sqrt{2}32​ (C) 2\sqrt{2}2​ (D) 3\sqrt{3}3​

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Q.If a parallelopiped is formed by planes drawn through the points $(5,8,10)$ and $(3,6,8)$ parallel to the coordinate planes, then the length of diagonal of the parallelopiped is
(A) $2\sqrt{3}$
(B) $3\sqrt{2}$
(C) $\sqrt{2}$
(D) $\sqrt{3}$

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