The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane. The reference point analogous to the origin of a Cartesian system is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth. Polar Graph Paper: A polar grid with several angles labeled in degrees.
Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics. In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.
This point is plotted on the grid in Figure. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry. When given a set of polar coordinates, we may need to convert them to rectangular coordinates.
The location of a point in the plane is given by its coordinates, a pair of numbers enclosed in parentheses: x , y. The first number x gives the point's horizontal position and the second number y gives its vertical position.
All positions are measured relative to a "central" point called the origin, whose coordinates are 0 , 0. For example, the point 5 , 2 is 5 units to the right of the origin and 2 units up, as shown in the figure. Negative coordinate numbers tell us to go left or down.
See the other points in the figure for examples. Write your answer Related questions. How can coordinate proof be used to prove two lines are parallel? Where do two distinct parallel lines intersect? How would you describe the properties of a cylinder? Which of these statements describe properties of parallelograms? Two distinct lines that do not intersect are parallel? In a coordinate plane the two nonvertical lines are parallel if and only if they have what slope?
What statements describe properties of parallelograms? What happens to a polygon when you add to the x-coordinate? If two distinct lines intersect?
Is a square parallel? Latitude and departures in land surveying? What is the need of parallel line algorithm? What are the properties of an isosceles trapezoid? Describe what happen to parallel light rays hitting in a concave mirror? Are vertical lines parralel? What are the properties of parallel line? What are the properties of kite?
Properties of trapezium? A sphere is the set of all points in space equidistant from a fixed point, the center of the sphere Figure , just as the set of all points in a plane that are equidistant from the center represents a circle. In a sphere, as in a circle, the distance from the center to a point on the sphere is called the radius. The equation of a circle is derived using the distance formula in two dimensions.
In the same way, the equation of a sphere is based on the three-dimensional formula for distance. The sphere with center and radius can be represented by the equation. This equation is known as the standard equation of a sphere. Find the standard equation of the sphere with center and point as shown in Figure. Use the distance formula to find the radius of the sphere:.
Find the standard equation of the sphere with center containing point. Let and and suppose line segment forms the diameter of a sphere Figure. Find the equation of the sphere. Since is a diameter of the sphere, we know the center of the sphere is the midpoint of Then,.
Furthermore, we know the radius of the sphere is half the length of the diameter. This gives. Then, the equation of the sphere is. Find the equation of the sphere with diameter where and. Describe the set of points that satisfies and graph the set.
We must have either or so the set of points forms the two planes and Figure. The set of points forms the two planes and. Hint One of the factors must be zero. Describe the set of points in three-dimensional space that satisfies and graph the set.
The x — and y -coordinates form a circle in the xy -plane of radius centered at Since there is no restriction on the z -coordinate, the three-dimensional result is a circular cylinder of radius centered on the line with The cylinder extends indefinitely in the z -direction Figure. Describe the set of points in three dimensional space that satisfies and graph the surface. A cylinder of radius 4 centered on the line with. Hint Think about what happens if you plot this equation in two dimensions in the xz -plane.
Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments arrows. With a three-dimensional vector, we use a three-dimensional arrow. Three-dimensional vectors can also be represented in component form. The notation is a natural extension of the two-dimensional case, representing a vector with the initial point at the origin, and terminal point The zero vector is So, for example, the three dimensional vector is represented by a directed line segment from point to point Figure.
Vector addition and scalar multiplication are defined analogously to the two-dimensional case. If and are vectors, and is a scalar, then. If then is written as and vector subtraction is defined by.
The standard unit vectors extend easily into three dimensions as well— and —and we use them in the same way we used the standard unit vectors in two dimensions.
Thus, we can represent a vector in in the following ways:. Let be the vector with initial point and terminal point as shown in Figure. Express in both component form and using standard unit vectors. In component form,. Let and Express in component form and in standard unit form.
Write in component form first. As described earlier, vectors in three dimensions behave in the same way as vectors in a plane. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space Figure. We have already seen how some of the algebraic properties of vectors, such as vector addition and scalar multiplication, can be extended to three dimensions.
Other properties can be extended in similar fashion. They are summarized here for our reference. Let and be vectors, and let be a scalar. Scalar multiplication:. Vector addition:. Vector subtraction:. Vector magnitude:. Unit vector in the direction of v: if. We have seen that vector addition in two dimensions satisfies the commutative, associative, and additive inverse properties.
These properties of vector operations are valid for three-dimensional vectors as well. Scalar multiplication of vectors satisfies the distributive property, and the zero vector acts as an additive identity. The proofs to verify these properties in three dimensions are straightforward extensions of the proofs in two dimensions.
Let and Figure. Find the following vectors. First, use scalar multiplication of each vector, then subtract: Write the equation for the magnitude of the vector, then use scalar multiplication: First, use scalar multiplication, then find the magnitude of the new vector.
Note that the result is the same as for part b. The procedure is the same in three dimensions: Let and Find a unit vector in the direction of. Start by writing in component form. A quarterback is standing on the football field preparing to throw a pass.
The quarterback throws the ball at a velocity of 60 mph toward the receiver at an upward angle of see the following figure. Write the initial velocity vector of the ball, in component form.
The first thing we want to do is find a vector in the same direction as the velocity vector of the ball.
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