The unit vector a_n normal (perpendicular) to a planar surface S_0 is directed a
ID: 2085871 • Letter: T
Question
The unit vector a_n normal (perpendicular) to a planar surface S_0 is directed away from the origin of the rectangular coordinate system. The direction cosines of a_n are given by (cos alpha_x, cos alpha_y, cos alpha_z). Let P(x, y, z) be an arbitrary point on the plane S which is parallel to S_0 and of perpendicular distance d from the origin. (a) Express the position vector r of P(x, y, z) in terms of its coordinates (x, y, z). (b) Explain what the direction cosines (cos alpha_x, cos alpha_y, cos alpha_z) of the unit normal a_n mean. Then show the equation of S is given by: f(x, y, z) = cos alpha_z middot x + cos alpha_y middot y + cos alpha_z middot z = d (c) Compute the gradient of S from its equation shown in (b) and compare it to the unit normal a_n of S_0.Explanation / Answer
Answer:-a) Postion vector of a point is the vector from the origin to that point. So for given point P(x, y, z) the position vector wil be:- r = (x-0)ax + (y-0)ay + (z-0)az
= xax + yay + za .
b) Direction cosine of a vector is the cosine of angle made between the vector and coordinate axes.
Since planer surface S0 and plane S are parallel hence normal vector an is perpendicular to both place. Also point P is on plane S with position vector r. So we can write-
(r - dan ).dan = 0 , this gives r.an = d
keeping the values we get xcosax + ycosay + zcosaz = d.
Note:- Angle of cos is in term of "alpha", I am unable to type that so I used "a".
c) Gradient of S is given by del(f) = cosaxi + cosayj + cosaz k. i.e after doing the partial differentiation of x wrt dx, y wrt dy and of z wrt dz.
Note:- All bold letters show vector notation.
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