Question

The equation of the plane is given by
The position vectors of the points and are given by and respectively
The foot of the perpendicular from to plane is
is the point on produced such that
Find
(iv) The equation of the plane in the form
(v) The exact length of projextion of the line on

Answer 1

Subject: Plane Equation

Jawaban:

Known as :

Question: i) Vector position of B

ii) Distance from A to phi

iii) Vector position of C

iv) Equation plane DBC

v) Exact lenght of projextion of the line BD on pi

Penjelasan dengan langkah-langkah:

To solve this problem, first you have to find the equation of the throught A and perpendicular to the plane pi.

To search for the equation of the line through A and perpendicular to the plane just to find the equation of the line through A and parallel to the normal vector of plane.

Then, this parametric line equation is :

x=t-2

y=2t+1

z=-2t

Because A to plane pi is perpendicular at the point of B, then surely B at the plane as well as the point of intersection between the lines with respect to the plane. Substitution x = t - 2, y = 2t + 1, z = -2t, then :

(t - 2) + (4t + 2) + 4t = 18

9t = 18

t = 2

And then the point of B is (0, 5, -4), then the vector position of B is 5j - 4k.

ii) Distance

Distance from A to pi can you find with using Pythagoras :

or, using lenght of vector AB :

AB = b - a = (0, 5, -4) - (-2, 1, 0)

AB = (2, 4, -4)

iii) Vector position of C

Because C at the BA with BA:AC=2:1, you cannot immediately search for point C with a line segment distribution formula, so in order to find it with the formula, you must change it to the ratio BC: CA first.

BA = 2AC

BC + CA = -2CA

BC = -3CA

BC:CA = -3:1

So, point C (-3, -1, 2) so that the vector position of C is -3i - j + 2k.

iv) Assume D as the edge point of the plane DBC, then :

DB = b - d = (0, 5, -4) - (2, 5, -3) = (-2, 0, -1)

DC = c - d = (-3, -1, 2) - (2, 5, -3) = (-5, -6, 5)

So, the plane of DBC is :

g: A(x - a) + B(y - b) + C(z - c) = 0

g: -6(x - 2) + 15(y - 5) + 12(z + 3) = 0 (bagi 3)

g: -2x + 4 + 5y - 25 + 4z + 12 = 0

g: -2x + 5y + 4z = 9

v) The projextion formula of BD at pi is :

so that :

I hope this helps, sorry if my english is not good and the answer is incorrect.