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3D Geometry Problems for Students

The document contains a series of mathematical problems related to three-dimensional geometry, specifically focusing on lines, their intersections, angles, and distances. Each question requires the application of vector equations and geometric principles to find relationships between given points in 3D space. The problems range from proving perpendicularity to finding the shortest distance between lines.

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rakshitmathur19
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49 views2 pages

3D Geometry Problems for Students

The document contains a series of mathematical problems related to three-dimensional geometry, specifically focusing on lines, their intersections, angles, and distances. Each question requires the application of vector equations and geometric principles to find relationships between given points in 3D space. The problems range from proving perpendicularity to finding the shortest distance between lines.

Uploaded by

rakshitmathur19
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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APEEJAY SCHOOL NOIDA

THREE DIMENSIONAL GEOMETRY

LEVEL-1

Q1.Show that the line joining the points A ( 7 , 8 , 6 ) and B ( 9 , 11 , 7) is


perpendicular to the line joining the points C ( 9 ,1,-6 ) and D( 7 , 2 ,-5 ).

Q2. For what values of p and q will the line joining points A ( 3, 2 ,5 ) and

B ( p , 5 ,0 ) be parallel to the line joining points C ( 1 , 3 ,q ) and D( 6 , 4 , -1 ).

Q3.Find the coordinates of the foot of the perpendicular drawn from the point

A( 1 ,8,4 ) to the line joining the points B ( 0 , -1 ,3 ) and C ( 2 , -3 ,-1 ).

Q4.Find the vector equation of the line which is parallel to the vector 2𝑖̂ - 𝑗̂ + 3𝑘̂

And which passes through the point ( 5 , -2 , 4 ). Also find its Cartesian
equation.

Q5 Find the vector and the Cartesian equation of a line which passes through
𝑥+3 𝑦−4 𝑧+8
the point ( -2 ,4 ,-5) and is parallel to the line given by = =
3 5 6

𝑥−1 𝑦−2 𝑧−3 𝑥−4 𝑦−1


Q6.Show that the lines = = & = =z intersect .Find also the
2 3 4 5 2
point of intersection.

Q7.Determine whether the lines :


𝑥−1 𝑦+1 𝑧 𝑥+1 𝑦−2
= = and = , z = 2 intersect or not .
2 −1 1 5 1

𝑥−5 𝑦−7 𝑧+3 𝑥−8 𝑦−4 𝑧−5


Q8.Show that the lines = = & = = intersect .Find also the
4 4 −5 7 1 3
point of intersection.
𝑥−1 𝑦+1 𝑧−1 𝑥−2 𝑦−1 𝑧+1
Q9.Show that the lines = = & = = do not intersect .
3 2 5 4 3 −2

Q10. Find the angle between the lines:

𝑟⃗= 3𝑖̂ +2 𝑗̂ -4𝑘̂ + λ (𝑖̂ +2 𝑗̂ + 2𝑘̂) and 𝑟⃗ = +5 𝑗̂ -2𝑘̂ + µ (3𝑖̂ +2 𝑗̂ + 6𝑘̂)

Q11. Find the angle between the lines:


𝑥+1 𝑦−2 𝑧−1 𝑥+3 𝑧−4
= = & = , y = -5
5 −2 2 −2 3

Q12. Find value of p so that the lines:


1−𝑥 7𝑦−14 𝑧−3 7−7𝑥 𝑦−5 6−𝑧
= = & = = are perpendicular to each other.
3 2𝑝 2 3𝑝 1 5

Q13. Find value of p so that the lines:


𝑥−1 𝑦−2 𝑧−3 𝑥−1 𝑦−1 𝑧−6
= = & = = are perpendicular to each other.
−3 2𝑝 2 3𝑝 1 −5

Q14.Find the shortest distance between the lines whose vector equations are

𝑟⃗ = 𝑖̂ +2 𝑗̂ -4𝑘̂ + λ (2𝑖̂ +3 𝑗̂ + 6𝑘̂) &𝑟⃗= 3𝑖̂ +3 𝑗̂ -5𝑘̂ + µ (−2𝑖̂ +3 𝑗̂ + 8𝑘̂)

Q15.Find the shortest distance between the lines whose vector equations are

𝑟⃗ = 𝑖̂ +2 𝑗̂ -4𝑘̂ + λ (2𝑖̂ +3 𝑗̂ + 6𝑘̂) &𝑟⃗= 3𝑖̂ +3 𝑗̂ -5𝑘̂ + µ (2𝑖̂ +3 𝑗̂ + 6𝑘̂ )

Q16.Find the vector equation of the line passing through the point (2,3,2 )and
parallel to the line 𝑟⃗ = -2 𝑖̂ +3 𝑗̂ + λ (2𝑖̂ -3 𝑗̂ + 6𝑘̂). And find the distance
between these lines .

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