Determine whether the following pair of lines intersect:

Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Advertisement

Three Dimensional Geometry

Short Answer Type

121. Find the shortest distance between the lines:

straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 15 space straight j with hat on top space plus space 9 space straight k with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space left parenthesis 2 space straight mu minus 1 right parenthesis space straight i with hat on top space plus space left parenthesis 1 plus straight mu right parenthesis space straight j with hat on top space plus space left parenthesis 9 minus 3 space straight mu right parenthesis space straight k with hat on top

76 Views

Long Answer Type

122. Find the shortest distance between the lines:

straight r with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top space plus space straight lambda space open parentheses 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses

and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses.

80 Views

123. Find the shortest distance between the lines:

straight r with rightwards arrow space on top equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top space plus space straight lambda space open parentheses 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses

and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses.

73 Views

124. Find the shortest distance between the lines:

straight r with rightwards arrow on top space equals space straight i with hat on top space minus space 7 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight lambda space open parentheses stack straight i space with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses

and

straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight mu space open parentheses negative straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses

74 Views

Advertisement

125. Find the shortest distance between the lines:

straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top space minus space straight mu open parentheses 2 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top close parentheses

77 Views

126. Find the shortest distance between the lines:

straight r with rightwards arrow on top space equals space left parenthesis 8 plus 3 space straight lambda right parenthesis space straight i with hat on top space minus space left parenthesis 9 plus 16 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 10 plus space 7 space straight lambda right parenthesis space straight k with hat on top

and

straight r with rightwards arrow on top space equals space 15 space straight i with hat on top space plus space 29 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 8 space straight j with hat on top space minus space 5 space straight k with hat on top close parentheses.

77 Views

Short Answer Type

127. Find the distance between the lines l₁, and l₂ given by

straight r with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space space 6 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 3 straight j with hat on top space minus space 5 space straight k with hat on top space plus space straight mu space left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis

84 Views

Long Answer Type

128. Find the shortest distance between the lines:

fraction numerator straight x plus 1 over denominator 7 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 6 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction space and space fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 7 over denominator 1 end fraction

89 Views

Advertisement

Advertisement

129. Determine whether the following pair of lines intersect:
$straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top right parenthesis$
and $straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight mu left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis$

The equations of two lines are

straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top right parenthesis

...(1)
and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight mu left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis

...(2)

Comparing these equations with

straight r with rightwards arrow on top space equals space stack straight a subscript 1 with rightwards arrow on top space plus space straight lambda stack straight b subscript 1 with rightwards arrow on top

and

straight r with rightwards arrow on top space equals stack straight a subscript 2 with rightwards arrow on top plus space straight mu space stack straight b subscript 2 with rightwards arrow on top comma space space we space get comma

stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top comma space space stack straight b subscript 1 with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight k with hat on top comma stack straight a subscript 2 with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top comma space space stack straight b subscript 2 with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top

Let S be points on line (1) with position vector

stack straight a subscript 1 with rightwards arrow on top

and T be point on limit (2) with position vector

stack straight a subscript 2 with rightwards arrow on top

so that

ST with rightwards arrow on top space equals space stack straight a subscript 2 with rightwards arrow on top space minus space stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top

stack straight b subscript 1 with rightwards arrow on top space cross times space stack straight b subscript 2 with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 2 0 1 row 1 1 cell negative 1 end cell end table close vertical bar space equals space left parenthesis 0 minus 1 right parenthesis space straight i with hat on top space minus space left parenthesis negative 2 minus 1 right parenthesis space straight j with hat on top space plus space left parenthesis 2 minus 0 right parenthesis space straight k with hat on top space equals space minus straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close vertical bar space equals space square root of 1 plus 9 plus 4 end root space equals space square root of 14

Let

PQ with rightwards arrow on top

be the S.D. vector between given lines. Therefore, it is parallel to

stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top.

If

straight n with rightwards arrow on top

is a unit vector along

PQ with rightwards arrow on top comma

then

straight n with rightwards arrow on top space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator 1 over denominator square root of 14 end fraction left parenthesis negative straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis

Now S.D. = Projection of

ST with rightwards arrow on top space space on space space PQ with rightwards arrow on top space equals space Projection space of space ST with rightwards arrow on top space on space straight n with rightwards arrow on top space equals space stack ST. with rightwards arrow on top space straight n with rightwards arrow on top

equals space straight i with hat on top. space fraction numerator 1 over denominator square root of 14 end fraction left parenthesis negative straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 14 end fraction left parenthesis negative 1 plus 0 plus 0 right parenthesis space equals space minus fraction numerator 1 over denominator square root of 14 end fraction

equals space fraction numerator 1 over denominator square root of 14 end fraction left parenthesis in space magnitude right parenthesis space not equal to 0

∴ given lines do not interesect.

72 Views

Advertisement

130. Determine whether the following pair of lines intersect:

straight r with space rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top space plus space straight lambda left parenthesis 3 space straight i with hat on top space minus space straight j with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space straight k with hat on top space plus space straight mu space left parenthesis 2 straight i with hat on top space plus space 3 space stack straight k right parenthesis with hat on top

73 Views

Advertisement