AnGeom Bridging Final

CIVIL ENGINEERING BOARD EXAMS PROBLEMS PHILIPPINES – November 6, 2020
POLAR COORDINATES AND EQUATIONS
To form a polar coordinate system, start with a fixed point and call it the pole or origin. From this point draw a half line, or ray (usually horizontal and to the
right) and call this line the polar axis. In this system, the location of a point is expressed by its distance r from a fixed point and its angle from a fixed line.
Sign convention
1. θ is positive for counterclockwise and negative for clockwise.

2. r is positive for laid offs at terminal side and negative for laid offs at prologation through O from the terminal side θ .
DISTANCE BETWEEN TWO POINTS
CONVERSION OF RECTANGULAR TO POLAR COORDINATES AND VICE VERSA.
POLAR CURVES AND RECTANGULAR CURVES

EXAMPLES:
1. Convert (-4 , 1.077) to rectangular coordinates. (1.077 radians)
SOLUTION:
2. Rewrite to rectangular form: r = 3/(1 – 2cos θ)

SOLUTION:
3. Convert ( -1 , √3) to polar coordinates.
SOLUTION:
4. Find the polar equation of the circle whose rectangular equation is x2 + y2 – 8x + 6y – 2 = 0 .

SOLUTION:
5. Transform to polar equation: xy = 5

SOLUTION:
𝑥 = cos 𝜃 , 𝑦 = sin 𝜃
1
cos 𝜃 sin 𝜃 = 5(1) ; sin 2𝜃 = 5(sin2 𝜃 + cos2 𝜃)
2
sin 2𝜃 = 10 sin2 𝜃 + 10 cos2 𝜃 → 𝐴𝑛𝑠.
6. Convert to rectangular coordinates: (-5 , 90° )

SOLUTION :
EXERCISES – Answer the following questions.
1. Transform to rectangular form: r2sin 2θ = 6. Ans. xy = 3

2. Transform into polar coordinates: (2, 2). Ans. (2√2 , π/4)
3. Convert to polar form: (x +1)2 + y2 = 1 Ans. r = - 2cos θ
4. Find the polar coordinates of R( - 8, -12) . Ans. (14.42 , 4.12)
5. Write the polar equation of r = 2 csc θ . Ans. y = 2
6. The polar equation of an ellipse is equal to r2(4 sin2 θ + 9 cos2 θ) = 36. Compute its area. Hint (A = πab) Ans. 6π
7. A surveyor identifies a landmark at the point with polar coordinates (325, 70°) . Find the rectangular coordinates. Ans. (111.157 , 305.4)
8. Find the distance between (9, -45°) and ( -4 , 70°) . Ans. 8.16
9. Convert to rectangular form: r(sin θ – 3 cos θ) = 2. Ans. 3x = y – 2
10. Convert to polar form : x2 + y2 = 25. Ans. r = ± 5
11. Convert to rectangular form: r = 4cos θ. Ans. x2 - 4x + y2 = 0
12. A surveyor mapping out the land where a new housing development will be built identifies a landmark 𝟐𝟐𝟑 feet away and 𝟒𝟓° left of center. A
second landmark is 𝟒𝟏𝟖 feet away and 𝟕𝟎° right of center. Find the distance between two points. Ans. 550 feet
13. Find the rectangular form of r = 4. Ans. x2 + y2 = 16
14. Rewrite the Cartesian equation x2 + y2 = 6y as a polar equation. Ans. r = 6 sin θ
15. Convert ( -4 , 2π/3) into Cartesian Coordinates. Ans. (2 , -2√3 )
SOLID ANALYTIC GEOMETRY
SPACE COORDINATE SYSTEM – There three coordinate system used in solid analytic geometry:
1. RECTANGULAR COORDINATES – a point P(x , y , z) in space is fixed by its three distance x, y, and z from the coordinate planes.
2. CYLINDRICAL COORDINATES – A point P in space may be imagined as being on the surface of a cylinder perpendicular to the XY- plane. P(r, θ
,z) is fixed by its distance z from the xy- plane and by the polar coordinates (r , θ) of the projection of P on the XY-plane.
3. SPHERICAL COORDINATES – A point P in space may be imagined as being on the surface of a sphere with center at the origin O and radius r.
P(r, ϕ , θ) is fixed by its distance r from O, the angle ϕ between OP and z-axis , and the angle θ which is the angle between the x axis and the
projection OP on the xy- plane.
DISTANCE BETWEEN TWO POINTS

CONVERSION FACTORS : FROM RECTANGULAR AND VICE VERSA.
TO CYLINDRICAL : TO SPHERICAL:
DISTANCE FROM A POINT TO A PLANE: GENERAL EQUATION OF THE PLANE
EQUATION OF THE PLANE (INTERCEPT FORM) ANGLE BETWEEN TWO PLANES
PERPENDICULAR DISTANCE BETWEEN TWO PLANES
POINT OF INTERSECTION OF THE MEDIANS

SPACE COORDINATE SYSTEM COSINE FORMULAS (More useful when we take three dimensional forces in Engineering Mechanics)
SOLID ANALYTIC GEOMETRY GRAPHS
EXAMPLES
1. Calculate the distance between A (0,2,0) and B(7,2,-1) .
SOLUTION:
2. What is the distance between the point P(1,2,3) and the plane 2x + 2y – 3y + 3 = 0?
SOLUTION:
3. Convert to rectangular coordinates: (4 , 2π/3 , -2)

SOLUTION:
4. Determine the angles of the radius vector of the point (3,-2,5) that forms with the coordinate axes.
SOLUTION:
5. Find the direction cosines of a point having a coordinates of (2, 3, -6).

SOLUTION:
6. Convert the point (-1 , 1, -√2) to spherical coordinates.

SOLUTION:
EXERCISES – Answer the following questions.
1. Convert from cylindrical to spherical coordinates: (1, π/2 , 1) (Hint: Convert first to rectangular form) Ans. (√2 , π/2 , π/4)
2. Find the distance from a point (4, -4, 3) to the plane 2x – 2y + 5z + 8 = 0. Ans. 6.8
3. If 1/2 , 1/√2 , cos γ are the direction cosines of the vector, find γ . Ans. No solution
4. A given sphere has the equation x2 + y2 + z2 + 4x – 6y – 10z + 13 = 0 . Find the radius. Ans. 5
5. Determine the direction cosines of the normal to the plane x + y + z = 1. Ans. cos α = cos β = cos γ = 1/√3
6. Calculate the distance of the planes 2x – y – 2z + 5 = 0 and 4x – 2y – 2z + 15 = 0. Ans. 0.833
7. Find the direction cosines of the line passing through points ( -2, 4, -5) and (1, 2, 3). Ans. cos α = 3/√77 , cos β = -2/√77 , cos γ = 8/√77
8. The vertices of a triangle are (1,1,0) , (1, 0 , 1) and ( 0, 1, 1) . Find the point of intersection of the medians of the triangle. Ans. (2/3 , 2/3, 2/3)
9. Find the midpoint of the points (5, 12, 10) and (3, 0 , -1). Ans. (4,6,4.5)
10. Find the angle between two planes x – 2y +z = 0 and 2x + 3y – 2z = 0. Ans. 126.448º
11. Find the distance between the given plane 2x + 4y – 4x – 6 = 0 and P(0,3,6). Ans. 3
12. Convert (1, -1 , -√2) to spherical form. Ans. (2, 7π/4 , 3π/4)
13. Find the equation of the sphere x2 + y2 + z2 = 4 in cylindrical form. Ans. r2 + z2 = 4
14. Determine the cos γ for the point (2 , 3 , 4) . Ans. 4/√29
15. Calculate the distance between points (6, 11, 3) and (4, 6, 12). Ans. 10.5
END OF ANALYTIC GEOMETRY
Next Topics on November 9, 2020 – Solid Mensuration
1. Squares and Rectangles

2. Parallelograms , Trapezoids and Other Quadrilaterals
REFERENCES
1. College Algebra with Trigonometry by Barnett

2. Engineering Mathematics by Gillesania
3. Algebra and Trigonometry by Openstax
4. College Algebra and Trigonometry by Cynthia Young
5. Analytic Geomtry by Burdette
6. https://www.lcps.org/cms/lib/VA01000195/Centricity/Domain/10387/8.1%20Worksheet%20Polar%20Coordinates%20KEY.pdf
7. Glencoe’s Advanced Mathematical Concepts
8. Engineering Mathematics Vol. 1 by Besavilla
9. https://www.ck12.org/book/ck-12-trigonometry-concepts/section/6.2/
10. https://math.libretexts.org/Bookshelves/Algebra/Book%3A_Algebra_and_Trigonometry_(OpenStax)/10%3A_Further_Applications_of_Trigonometr
y/10.05%3A_Polar_Coordinates_-_Graphs
11. https://precalculuscoach.com/wp-content/uploads/2017/11/9-1-Word-Problems-Polar-Coordinates.pdf
12. https://tutorial.math.lamar.edu/classes/calcii/polarcoordinates.aspx
13. Analytic Geometry by Rainville
14. https://www.sangakoo.com/en/unit/distance-between-two-points-in-space
15. http://mathonline.wikidot.com/the-distance-between-a-plane-and-a-point
16. https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.7%3A_Cylindrical_and_Spherica
l_Coordinates
17. https://tutorial.math.lamar.edu/classes/calciii/CylindricalCoords.aspx
18. https://tutorial.math.lamar.edu/classes/calciii/SphericalCoords.aspx
19. https://www.math.utah.edu/lectures/math2210/9PostNotes.pdf
20. https://www.onlinemath4all.com/direction-cosines-word-problems-in-vectors.html
21. https://www.superprof.co.uk/resources/academic/maths/analytical-geometry/distance/distance-between-a-point-and-a-plane.html
22. https://www.teachoo.com/3559/756/Example-3----Find-direction-cosines-of-line-passing-through/category/Examples/
23. https://math.stackexchange.com/questions/1528909/find-the-angle-between-two-planes-using-their-normal-vectors
24. https://mosaic.math.tamu.edu/~glahodny/Math251/Section%2013.9.pdf
25. https://www.whitman.edu/mathematics/calculus_late_online/section14.06.html
26. https://slideplayer.com/slide/13460064/
27.

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CIVIL ENGINEERING BOARD EXAMS PROBLEMS PHILIPPINES – November 6, 2020

POLAR COORDINATES AND EQUATIONS

1. θ is positive for counterclockwise and negative for clockwise.

DISTANCE BETWEEN TWO POINTS

CONVERSION OF RECTANGULAR TO POLAR COORDINATES AND VICE VERSA.

POLAR CURVES AND RECTANGULAR CURVES

2. Rewrite to rectangular form: r = 3/(1 – 2cos θ)

4. Find the polar equation of the circle whose rectangular equation is x2 + y2 – 8x + 6y – 2 = 0 .

5. Transform to polar equation: xy = 5

sin 2𝜃 = 10 sin2 𝜃 + 10 cos2 𝜃 → 𝐴𝑛𝑠.

6. Convert to rectangular coordinates: (-5 , 90° )

EXERCISES – Answer the following questions.

1. Transform to rectangular form: r2sin 2θ = 6. Ans. xy = 3

DISTANCE BETWEEN TWO POINTS

DISTANCE FROM A POINT TO A PLANE: GENERAL EQUATION OF THE PLANE

EQUATION OF THE PLANE (INTERCEPT FORM) ANGLE BETWEEN TWO PLANES

PERPENDICULAR DISTANCE BETWEEN TWO PLANES

POINT OF INTERSECTION OF THE MEDIANS

SOLID ANALYTIC GEOMETRY GRAPHS

3. Convert to rectangular coordinates: (4 , 2π/3 , -2)

5. Find the direction cosines of a point having a coordinates of (2, 3, -6).

6. Convert the point (-1 , 1, -√2) to spherical coordinates.

END OF ANALYTIC GEOMETRY

Next Topics on November 9, 2020 – Solid Mensuration

1. Squares and Rectangles

1. College Algebra with Trigonometry by Barnett

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