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 θ between them is given by
cos θ = 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}(√2 / 3)
Straight Line in Space
The two equations of the line ax + by + cz + d = 0 and a’ x + b’ y + c’ z + d’ = 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 – x_{1} / a = y – y_{1} / b = z – 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} + λa, y_{1} + λb, z_{1} + λc), where λ ∈ R is parameter. If a, b, c are replaced by direction cosines 1, m, n, then λ, 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 – x_{1} / x_{2} – x_{1} = y – y_{1} / y_{2} – y_{1} = z – z_{1} / z_{2} – z_{1}
3. Vector equation of a line passing through a point with position vector a and parallel to vector b is r = a + λ 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 + λ (b – a) , where λ is a parameter.
5. (a) The length of the perpendicular from a point on the line r – a + λ 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} + λb_{1} and r = a_{2} + μb_{2} is given by
10. The shortest distance parallel lines r = a_{1} + λb_{1} and r = a_{2} + μb_{2} is given by
11. Lines r = a_{1} + λb_{1} and r = a_{2} + μb_{2} are intersecting lines, if (b_{1} * b_{2}) * (a_{2} – 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 – x_{1} / a = y – y_{1} / b = z – z_{1} / c = – 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 – x_{1} / a = y – y_{1} / b = z – z_{1} / c = – (ax_{1} + by_{1} + cz_{1} + d) / a^{2} + b^{2} + c^{2}
14. Since, x, y and zaxes pass through the origin and have direction cosines (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. Therefore, their equations are
x – axis : x – 0 / 1 = y – 0 / 0 = z – 0 / 0
y – axis : x – 0 / 0 = y – 0 / 1 = z – 0 / 0
z – axis : x – 0 / 0 = y – 0 / 0 = z – 0 / 1
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1. What is the angle between two intersecting lines? 
2. How can the angle between two intersecting lines be calculated? 
3. What is the shortest distance between two intersecting lines? 
4. How can the shortest distance between two intersecting lines be determined? 
5. Can the angle between two intersecting lines be greater than 180 degrees? 

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