Egypt University of Informatics
FACULTY OF Engineering
Fall 2022
0
10
Assignment (1)
Due Date: 28 /10 /2022
PHM111: Multivariate Calculus
Name:
ID:
Section (12.5):
Q.1
Determine whether the lines 𝐿1 and 𝐿2 are parallel, skew, or intersecting. If they intersect,
find the point of intersection.
𝐿1 :
𝐿2 :
𝑥−2
1
𝑥−3
1
=
=
𝑦−3
−2
𝑦+4
3
=
=
𝑧−1
−3
𝑧−2
−7
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Q.2
Find the equation of the plane that passes through the line of intersection of the planes
𝑥 − 𝑧 = 1 and 𝑦 + 2𝑧 = 3 is perpendicular to the plane 𝑥 + 𝑦 − 2𝑧 = 1.
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ID.:
Q.3
Find the point at which the line intersects the given plane:
𝑦
5𝑥 = = 𝑧 + 2; 10𝑥 − 7𝑦 + 3𝑧 + 24 = 0
2
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Section (12.6):
Q.4
Reduce the equation to one of the standard forms, classify the surface, and sketch it.
𝑥 2 − 𝑦 2 − 𝑧 2 − 4𝑥 − 2𝑧 + 3 = 0
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ID.:
Q.5
Find an equation for the surface obtained by rotating the line 𝑧 = 2𝑦 about the z- axis.
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Section (13.1):
Q.6
Find the domain of the vector function.
𝑡2 − 𝑡
sin 𝜋𝑡
lim (
𝑖 + √𝑡 + 8 𝑗 +
𝑘)
𝑡→1 𝑡 − 1
ln 𝑡
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ID.:
Q.7
Sketch the curve with the given vector equation. Indicate with an arrow the direction in
which 𝑡 increases
𝑟(𝑡) = ⟨sin 𝜋𝑡, 𝑡, cos 𝜋𝑡⟩
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Q.8
Show that the curve with parametric equations 𝑥 = sin 𝑡, 𝑦 = cos 𝑡, 𝑧 = sin2 𝑡 is the
curve of intersection of the surfaces 𝑧 = 𝑥 2 and 𝑥 2 + 𝑦 2 = 1. Use this fact to help sketch
the curve.
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Q.9
At what points does the helix 𝑟(𝑡) = ⟨sin 𝑡, cos 𝑡, 𝑡⟩ intersect the sphere
𝑥 2 + 𝑦 2 + 𝑧 2 = 5?
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Section (13.2):
Q.10
Find a vector equation for the tangent line to the curve of intersection of the cylinders
𝑥 2 + 𝑦 2 = 25 and 𝑦 2 + 𝑧 2 = 20 at the point (3,4,2).
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ID.:
Q.11
Find parametric equations for the tangent line to the curve with the given parametric
equations at the specified point:
𝑥 = 𝑡,
𝑦 = 𝑒 −𝑡 ,
𝑧 = 2𝑡 − 𝑡 2 ; (0, 1, 0)
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Section (13.3):
Q.12
Find the length of the curve correct to four decimal places:
𝑟(𝑡) = ⟨cos 𝜋𝑡, 2𝑡 , sin 2𝜋𝑡⟩, from (1, 0, 0) to (1, 4, 0).
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