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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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The equation of plane is

straight x plus 2 straight y minus 2 straight z space equals space 9 space space space space or space space space straight x over 9 plus fraction numerator 2 straight y over denominator 9 end fraction space minus space fraction numerator 2 straight z over denominator 9 end fraction space equals space 1

straight x over 9 plus space fraction numerator straight y over denominator begin display style 9 over 2 end style end fraction plus fraction numerator straight z over denominator negative begin display style 9 over 2 end style end fraction space equals space 1

which is of intercept form

straight x over straight a plus straight y over straight b plus straight z over straight c equals 1

therefore space space straight a space equals space 9 comma space space space straight b space equals space 9 over 2 comma space space straight c space equals space minus 9 over 2 therefore space space space space intercepts space are space 9 comma space 9 over 2 comma space minus 9 over 2.

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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.

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139. Find the value of k for which the planes 3 x– 6 y– 2 z = 7 and 2x + y'– k z = 5 are perpendicular to each other.

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140.

The foot of the perpendicular drawn from the origin to the plane is (4, 3, 2). Find the equation of the plane.

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