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Exercise No. 8: Vectors and Geometry in Space

This document contains 15 questions about vectors and geometry in 3D space. It covers topics such as describing sets of points defined by equations in 3D space, finding distances and directions between points, working with vectors including finding dot and cross products, and calculating areas of shapes defined by points in 3D such as triangles, parallelograms, and parallelepipeds. The questions progress from simpler concepts involving individual points and vectors to more complex problems involving multiple vectors and geometric shapes in 3D space.

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Mohib Ullah
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119 views4 pages

Exercise No. 8: Vectors and Geometry in Space

This document contains 15 questions about vectors and geometry in 3D space. It covers topics such as describing sets of points defined by equations in 3D space, finding distances and directions between points, working with vectors including finding dot and cross products, and calculating areas of shapes defined by points in 3D such as triangles, parallelograms, and parallelepipeds. The questions progress from simpler concepts involving individual points and vectors to more complex problems involving multiple vectors and geometric shapes in 3D space.

Uploaded by

Mohib Ullah
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Exercise No.

8
Exercise No 12.1, 12.2, 12.3, 12.4
Vectors and Geometry in Space
Dr. Dur-e-Shehwar Sagheer

Question No. 1. (Sets, Equations and Inequalities)

Give geometric description of the set of points in space whose coordinates satisfy the
following equations/inequalities:

1. x = 2, y = 3

2. y = 0, z = 0

3. x2 + y 2 = 4, z=0

4. x2 + y 2 + z 2 = 1 x = 0

5. x2 + y 2 + z 2 = 25 y = −4

6. x2 + y 2 + z 2 = 1 x = 0

7. x2 + z 2 = 4, y=0

8. x2 + y 2 + (z + 3)2 = 25 z = 0

9. (a) x ≥ 0, y ≥ 0, z = 0

(b) x ≥ 0, y ≤ 0, z = 0

10. (a) x2 + y 2 + z 2 ≤ 1

(b) x2 + y 2 + z 2 > 1

11. (a) x2 + y 2 + z 2 = 1, z≥0

(b) x2 + y 2 + z 2 ≤ 1, z≥0

Question No. 2.

Describe the given set with single equation or with pair of equations:

1. (a) x − axis at (3, 0, 0)


(b) y − axis at (0, −1, 0)
(c) z − axis at (0, 0, −2)

1
2. The plane through the point (3, −1, 1) parallel to the

(a) xy-plane
(b) yz-plane
(c) xz-plane

3. Circle of radius 2 centered at (0, 2, 0) and lying in the

(a) xy-plane
(b) yz-plane
(c) xz-plane

Question No. 3.
Find the distance between points:

1. P1 (1, 1, 1), P2 (3, 3, 0)

2. P1 (1, 4, 5), P2 (4, −2, )

Question No. 4.

Find the center and radius of the sphere:

1. 2x2 + 2y 2 + 2z 2 + x + y + z = 9

2. x2 + y 2 + z 2 − 6y + 8z = 0

Question No. 5.

Find the equation of the sphere:


√
1. Center=(1, 2, 3) and radius = 13

2
Vectors in Space
Question No. 6.

Express the following vectors as product of its length and direction:

1. 2i + j − 2k
1 1 1
2. √ i − √ j − √ k
6 6 6
Question No. 7.

Find the vectors whose lengths and direction is given:


Length Direction
√ −3 4
2 i− k
5 5
6 2 3
7 i− j+ k
7 7 7
Question No. 7.

Find the directions of P1~P2 and mid point of the segment P1 P2 :

P1 (−1, 1, 5), P2 (2, 5, 0)


Question No. 8.
For the following vectors find

(a) u · v, |u|, |v|,

(b) the cosine of the angle between u and v,

(c) the scalar component of u in direction of v,

(d) projv u

1. v = 10i + 11j − 11k, u = 3i + 4k

2. v = 5j − 3k, u = i + j + k
   
1 1 1 −1
3. v = √ , √ , v= √ , √
2 3 2 3

Question No. 9.

Find the angles between the following vectors to the nearest hundredth of a radian:

1. u = 2i + j, v = i + 2j − k
√ √
2. u = 3i − 7j, v = 3i + j + 4k

3
Question No. 10.

Find the measures of angles of a triangle whose vertices are A(−1, 0), B(2, 1), C(1, −2).
Question No. 11.
Find length and direction of u × v and v × u.
1. u = 2i − 2j − k, v=i−j
2. u = 2i v = −3j
3. u = −8i − 2j − 4k, v = 2i + 2j + k
Question No. 12.
Sketch the coordinate axis and include u, v and u × v as vectors straight at the origin
1. u = i, v=k
2. u = i + j v = i − j
Question No. 13.
Find area of triangle determined by the points P, Q and R . Also find the unit vector
perpendicular to plane P QR.
1. P (1, −1, 2) Q(2, 0, −1) R(0, 2, 1)
2. P (2, −2, 1) Q(2, 1, 3) R(3, −1, 1)
Question No. 14.
Verify that (u × v) · w = (v × w) · u = (w × u) · v and find the volume of parallelepiped
(box) determined by u, v, and w.
1. u = 2i v = 2j w = 2k
2. u = 2i + j v = 2i − j + k, w = i + 2k
Question No. 15.
Let u = 5i − j + k v = j − 5k w = −15i + 3j + 3k. Which vectors if any are
a) perpendicular?
b) parallel? Give reasons for your answer.
Question No. 15.
Find the area of the parallelegram whose vertices are:
1. A(1, 0), B(0, 1) C(−1, 0), D(0, −1)
2. A(−1, 2), B(2, 0) C(7, 1), D(4, 3)
Question No. 15.
Find the area of the triangle whose vertices are:
1. A(0, 0), B(−2, 3) C(3, 1)
2. A(−5, 3), B(1, −2) C(6, −2).

******GOOD LUCK******

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