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Three Dimensional Geometry

Long Answer Type

131. Determine whether the following pair of lines intersect:

fraction numerator straight x minus 1 over denominator 2 end fraction space equals fraction numerator straight y plus 1 over denominator 3 end fraction space equals straight z

fraction numerator straight x plus 5 over denominator 5 end fraction space equals space fraction numerator straight y minus 2 over denominator 1 end fraction comma space space straight z space equals space 2

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Short Answer Type

132. Find the intercepts cut off by the plane x + 2y – 2z = 9 with the axes.

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133. Find the equation of the plane with intercepts 2, 3 and 4 on the x, y and z-axis respectively.

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134. Find the intercepts cut off by the plane 2x + y – z = 5.

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135. Find the intercept form of the equation of the plane x + 3 y – 7 z + 2 = 0.

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136. Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane..

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137. Find a unit normal vector to the plane x + 2y + 3z – 6 = 0.

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138. Find the normal form of the equation of the plane 3x– 4y + z + 5 = 0.

The equation of plane is 3x – 4y + z + 5 = 0
or 3 x – 4 y + z = – 5
or – 3 x + 4 y – z = 5
Dividing both sides by $square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared end root space equals space square root of 26 comma space space we space get$
$negative fraction numerator 3 over denominator square root of 26 end fraction straight x plus fraction numerator 4 over denominator square root of 26 end fraction straight y space minus space fraction numerator 1 over denominator square root of 26 end fraction straight z space equals space fraction numerator 5 over denominator square root of 26 end fraction$
which is of form lx + my + nz = p
where $straight l equals fraction numerator negative 3 over denominator square root of 26 end fraction comma space space space straight m space equals space fraction numerator 4 over denominator square root of 26 end fraction comma space space straight n space equals negative fraction numerator 1 over denominator square root of 26 end fraction comma space space straight p space equals space fraction numerator 5 over denominator square root of 26 end fraction$
The direction-cosines of the normal to the plane are $fraction numerator negative 3 over denominator square root of 26 end fraction comma space fraction numerator 4 over denominator square root of 26 end fraction comma space minus fraction numerator 1 over denominator square root of 26 end fraction.$