Mathematics Lab Activity-13 Class XII

Mathematics Laboratory Activities 12 on Three Dimensional Geometry for class XI Non-Medical students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 11

three dimensional geometry

Activity - 13

Objective

To measure the shortest distance between two skew lines and verify it analytically.

Material Required

A piece of plywood of size 30cm x 20 cm, a squared paper, three wooden blocks of size 2cm x 2cm x 2cm each and one wooden block of size 2 cm x 2 cm x 4cm, wires of different lengths, set squares, adhesive, pen/pencils, etc.

Procedure

1. Paste a squared paper on a piece of plywood.

2. On a squared paper, draw two lines OA and OB to represent x- axis, and y-axis, respectively.

3. Name the three blocks of size 2cm x 2cm x 2cm as I, II and III, name the other wooden block of size 2 cm x 2 cm x 4cm as IV.

4. Place blocks I, II, III such that their base centres are at the points (2, 2), (1, 6) and (7, 6), respectively, and block IV with its base centre at (6, 2).

5. Place a wire joining the points P and Q, the centres of the bases of the blocks I and III and another wire joining the centres R and S of the tops of blocks II and IV as shown in figure 13.1. These two wires represent two skew lines as these are connected in different planes.

6. Take a wire and join it perpendicularly with the skew lines and measure the actual distance.

Figure 13.1

7. A set square is placed in such a way that its one perpendicular side is along the wire PQ.

8. Move the set-square along PQ till its other perpendicular side touches the other wire.

9. Measure the distance between the two lines in this position using set-square. This is the shortest distance between two skew lines.

10. Analytically, find the equation of line joining P(2, 2, 0) and Q(7, 6, 0) and other line joining R(1, 6, 2) and S(6, 2, 4) and find S.D. using the formula

$d=\left|\frac{\left(\overrightarrow{a_{2}}-\overrightarrow{a_{1}}\right).\left(\overrightarrow{b_{1}}\times\overrightarrow{b_{2}}\right)}{\left|\overrightarrow{b_{1}}\times\overrightarrow{b_{2}}\right|}\right|$ .

The distance obtained in two cases will be the same.

Observations

1. Coordinates of point P are (2, 2, 0)

2. Coordinates of point Q are (7, 6, 0)

3. Coordinates of point R are (1, 6, 2)

4. Coordinates of point S are (6, 2, 4)

5. Equation of the line PQ is = $\frac{x-2}{5}=\frac{y-2}{4}=\frac{z}{0}=\lambda$

6. Equation of the line RS is = $\frac{x-1}{5}=\frac{y-6}{-4}=\frac{z-2}{2}=\mu$

7. Shortest distance between PQ and RS analytically = 3.0476 cm

Shortest distance by actual measurement = 3 cm

Result

Here we conclude that the shortest distance between two lines calculated analytically and by actual measurement is approximately same.

Applications

This activity can be used to explain the concept of skew lines and of shortest distance between two lines in the space.

VIVA – VOICE

Q1. What are skew lines ?
Ans. Skew lines are those lines which are neither intersecting nor parallel.

Q2. What is shortest distance between teo lines ?
Ans. It is that distance which is shortest to all the distances between two points lying on both lines.

Q3. What is the shortest distance between two intersecting lines ?
Ans. Zero.

Q4. Is shortest distance between two lines perpendicular to both the lines ?
Ans. Yes

Q5. Write down example of two skew lines ?
Ans. A pair of lines through opposite edges of a regular tetrahedron.

Q6. Write down shortest distance between two parallel lines $\vec{r}=\vec{a_{1}}+\lambda\vec{b}$ and $\vec{r}=\vec{a_{2}}+\lambda\vec{b}$ .
Ans.
$d=\frac{|\left(\vec{a_{2}}-\vec{a_{1}}\right)\times\vec{b}|}{|\vec{b}|}$
Q7. Write down formula for shortest distance between lines $\vec{r}=\vec{a_{1}}+\lambda\vec{b_{1}}$ and $\vec{r}=\vec{a_{2}}+\lambda\vec{b_{2}}$ .
Ans.

$d=\frac{|\left(\vec{a_{2}}-\vec{a_{1}}\right).(\vec{b_{1}}\times\vec{b_{2}})|}{|\vec{b_{1}}\times\vec{b_{2}}|}$

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