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**Angle Between Two Intersecting Lines**

If l(x_{1}, m_{1}, n_{1}) and l(x_{2}, m_{2}, n_{2}) be the direction cosines of two given lines, then the angle Î¸ between them is given by

cos Î¸ = l_{1}1_{2} + m_{1}m_{2} + n_{1}n_{2}

(i) The angle between any two diagonals of a cube is cos^{-1} (1 / 3).

(ii) The angle between a diagonal of a cube and the diagonal of a face (of the cube is cos^{-1}(âˆš2 / 3)

**Straight Line in Space**

The two equations of the line ax + by + cz + d = 0 and aâ€™ x + bâ€™ y + câ€™ z + dâ€™ = 0 together represents a straight line.

1. Equation of a straight line passing through a fixed point A(x_{1}, y_{1}, z_{1}) and having direction ratios a, b, c is given by

x â€“ x_{1} / a = y â€“ y_{1} / b = z â€“ z_{1} / c, it is also called the symmetrically form of a line.

Any point P on this line may be taken as (x_{1} + Î»a, y_{1} + Î»b, z_{1} + Î»c), where Î» âˆˆ R is parameter. If a, b, c are replaced by direction cosines 1, m, n, then Î», represents distance of the point P from the fixed point A.

2. Equation of a straight line joining two fixed points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) is given by

x â€“ x_{1} / x_{2} â€“ x_{1} = y â€“ y_{1} / y_{2} â€“ y_{1} = z â€“ z_{1} / z_{2} â€“ z_{1}

3. Vector equation of a line passing through a point with position vector a and parallel to vector b is r = a + Î» b, where A, is a parameter.

4. Vector equation of a line passing through two given points having position vectors a and b is r = a + Î» (b â€“ a) , where Î» is a parameter.

5. (a) The length of the perpendicular from a point on the line r â€“ a + Î» b is given by

(b) The length of the perpendicular from a point P(x_{1}, y_{1}, z_{1}) on the line

where, 1, m, n are direction cosines of the line.

6. **Skew Lines** Two straight lines in space are said to be skew lines, if they are neither parallel nor intersecting.

7. **Shortest Distance** If l_{1} and l_{2} are two skew lines, then a line perpendicular to each of lines 4 and 12 is known as the line of shortest distance.

If the line of shortest distance intersects the lines l_{1} and l_{2} at P and Q respectively, then the distance PQ between points P and Q is known as the shortest distance between l_{1} and l_{2}.

8. The shortest distance between the lines

9. The shortest distance between lines r = a_{1} + Î»b_{1} and r = a_{2} + Î¼b_{2} is given by

10. The shortest distance parallel lines r = a_{1} + Î»b_{1} and r = a_{2} + Î¼b_{2} is given by

11. Lines r = a_{1} + Î»b_{1} and r = a_{2} + Î¼b_{2} are intersecting lines, if (b_{1} * b_{2}) * (a_{2} â€“ a_{1}) = 0.

12. The image or reflection (x, y, z) of a point (x_{1}, y_{1}, z_{1}) in a plane ax + by + cz + d = 0 is given by

x â€“ x_{1} / a = y â€“ y_{1} / b = z â€“ z_{1} / c = â€“ 2 (ax_{1} + by_{1} + cz_{1} + d) / a^{2} + b^{2} + c^{2}

13. The foot (x, y, z) of a point (x_{1}, y_{1}, z_{1}) in a plane ax + by + cz + d = 0 is given by

x â€“ x_{1} / a = y â€“ y_{1} / b = z â€“ z_{1} / c = â€“ (ax_{1} + by_{1} + cz_{1} + d) / a^{2} + b^{2} + c^{2}

14. Since, x, y and z-axes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. Therefore, their equations are

x â€“ axis : x â€“ 0 / 1 = y â€“ 0 / 0 = z â€“ 0 / 0

y â€“ axis : x â€“ 0 / 0 = y â€“ 0 / 1 = z â€“ 0 / 0

z â€“ axis : x â€“ 0 / 0 = y â€“ 0 / 0 = z â€“ 0 / 1

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