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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
- θ is positive for counterclockwise and negative for clockwise.
- 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:
- Convert (-4 , 1.077) to rectangular coordinates. (1.077 radians)
SOLUTION:
- Rewrite to rectangular form: r = 3/(1 – 2cos θ)
SOLUTION:
SOLID ANALYTIC GEOMETRY
SPACE COORDINATE SYSTEM – There three coordinate system used in solid analytic geometry:
- RECTANGULAR COORDINATES – a point P(x , y , z) in space is fixed by its three distance x, y, and z from the coordinate planes.
- 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.
- 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
- What is the distance between the point P(1,2,3) and the plane 2x + 2y – 3y + 3 = 0?
SOLUTION:
- Convert to rectangular coordinates: (4 , 2π/3 , - 2)
SOLUTION:
- Determine the angles of the radius vector of the point (3,-2,5) that forms with the coordinate axes.
SOLUTION:
- Find the direction cosines of a point having a coordinates of (2, 3, - 6).
SOLUTION:
- Convert the point (-1 , 1, - √2) to spherical coordinates.
SOLUTION:
EXERCISES – Answer the following questions.
- Convert from cylindrical to spherical coordinates: (1, π/2 , 1) (Hint: Convert first to rectangular form) Ans. (√2 , π/2 , π/4)
- Find the distance from a point (4, - 4, 3) to the plane 2x – 2y + 5z + 8 = 0. Ans. 6.
- If 1/2 , 1/√ 2 , cos γ are the direction cosines of the vector, find γ. Ans. No solution
- A given sphere has the equation x
2
2
2
- 4x – 6y – 10 z + 13 = 0. Find the radius. Ans. 5
- Determine the direction cosines of the normal to the plane x + y + z = 1. Ans. cos α = cos β = cos γ = 1/√ 3
- Calculate the distance of the planes 2x – y – 2z + 5 = 0 and 4x – 2y – 2z + 15 = 0. Ans. 0.
- 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
- 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)
- Find the midpoint of the points (5, 12, 10) and (3, 0 , - 1). Ans. (4,6,4.5)
- Find the angle between two planes x – 2y +z = 0 and 2x + 3y – 2z = 0. Ans. 126.448º
- Find the distance between the given plane 2x + 4y – 4x – 6 = 0 and P(0,3,6). Ans. 3
- Convert (1, - 1 , - √2) to spherical form. Ans. (2, 7 π/4 , 3π/4)
- Find the equation of the sphere x
2
2
2
= 4 in cylindrical form. Ans. r
2
2
- Determine the cos γ for the point (2 , 3 , 4). Ans. 4/√ 29
- 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
- Squares and Rectangles
- Parallelograms , Trapezoids and Other Quadrilaterals
