Class 12

Math

3D Geometry

Three Dimensional Geometry

A line segment has length 63 and direction ratios are $3,−2,6.$ The components of the line vector are

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From the point $P(1,2,4)$ a perpendicular is drawn on the plane $2x+y−2z+3=0$. Find the equation, the length and the coordinates of the foot of the perpendicular.

A line passes through the point $A(5,−2,4)$ and it is parallel to the vector $(2i^−j^ +3k^ )$. The vector equation of the line is

Write the vector equation of each of the following lines and hence determine the distance between them:$2x−1 =3y−2 =6z+4 $ and $4x−3 =6y−3 =12z+5 $.

Find the length and the foot of the perpendicular drawn from the point $(1,1,2)$ to the plane $r⋅(2i^−2j^ +4k^)+5=0$.

The Cartesian equations of a line are $3x+1=6y−2=1−z$. Find the fixed point through which it passes, its direction ratios and also its vector equation.

A line is perpendicular to two lines having direction ratios $1,−2,−2$ and $0,2,1$. The direction cosines of the line are

Show that the points $A(2,1,3),B(5,0,5)$ and $C(−4,3,−1)$ are collinear.

If the equations of a line are $−33−x =−2y+2 =6z+2 $, find the direction cosines of a line parallel to the given line.