direction cosine explaination
Answers
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction. Direction cosines are an analogous extension of the usual notion of slope to higher dimensions.
Answer:
Step-by-step explanation:
When a directed line OP passing through the origin makes α, β and γ angles with the x, y and z axis respectively with O as the reference, these angles are referred as the direction angles of the line and the cosine of these angles give us the direction cosines. These direction cosines are usually represented as l, m and n.
If we extend the line OP on the three dimensional Cartesian coordinate system, then to figure out the direction cosines, we need to take the supplement of the direction angles. It is pretty obvious from this statement that on reversal of the line OP in opposite direction, the direction cosines of the line also get reversed. In a situation where the given line does not pass through the origin, a line parallel to the given line passing through the origin is drawn and in doing so, the angles remain same as the angles made by the original line. Hence, we get the same direction.
Since we are considering a line passing through the origin to figure out the direction angles and their cosines, we can consider the position vectors of the line OP.
If = r, then from the above figure 1, we can see that
x= rcosα
y = rcosβ
z = rcosγ
Where r denotes the magnitude of the vector and it is given by,
r = (x–0)2+(y–0)2+(z–0)2−−−−−−−−−−−−−−−−−−−−√
⇒r =x2+y2+z2−−−−−−−−−−√
The cosines of direction angles are given by cosα, cosβ and cosγ and these are denoted by l, m and n respectively. Therefore, the above equations can be reframed as:
x = rcosα =lr ——————————————————— (1)
y =rcosβ =mr——————————————————– (2)
z = rcosγ = nr——————————————————— (3)
