NCERT Class 12-Mathematics: Chapter –11 Three Dimensional Geometry Part 8

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Question 13:

If a variable line in two adjacent positions has direction cosines l and , show that the small angle between the two positions is given by

Answer:

We have as direction cosines of a variable line in two different positions.

And

Now,

Question 14:

is the origin and is .Find the direction cosines of the line and the equation of plane through at right angle to .

Answer:

Question 15:

Two systems of rectangular axis have the same origin. If a plane cuts them at distances and , respectively, from the origin, prove that

Answer:

Consider are two system of rectangular axes.

Let their corresponding equation of plane be

And

Also, the length of perpendicular from origin to Eq. (i) and (ii) must be same.

Long Answer (L.A)

Question 16:

Find the foot of perpendicular from the point to the line . Also, find the perpendicular distance from the given point to the line.

Answer:

Question 17:

Find the distance of a point from the line

Answer:

We have equation of the line as

Let the coordinates of L be , then Dr’s of PL are .

Also, the direction ratios of given line are proportional to

Since, PL is perpendicular to the given line.

So, the coordinates of L are