Vectors
Coordinate Systems and Components of a Vector
Learning Objectives
By the end of this section, you will be able to:
- Describe vectors in two and three dimensions in terms of their components, using unit vectors along the axes.
- Distinguish between the vector components of a vector and the scalar components of a vector.
- Explain how the magnitude of a vector is defined in terms of the components of a vector.
- Identify the direction angle of a vector in a plane.
- Explain the connection between polar coordinates and Cartesian coordinates in a plane.
Vectors are usually described in terms of their components in a coordinate system. Even in everyday life we naturally invoke the concept of orthogonal projections in a rectangular coordinate system. For example, if you ask someone for directions to a particular location, you will more likely be told to go 40 km east and 30 km north than 50 km in the direction
In a rectangular (Cartesian) xy-coordinate system in a plane, a point in a plane is described by a pair of coordinates (x, y). In a similar fashion, a vector
As illustrated in (Figure), vector
It is customary to denote the positive direction on the x-axis by the unit vector
The vectors
If we know the coordinates
Displacement of a Mouse Pointer
A mouse pointer on the display monitor of a computer at its initial position is at point (6.0 cm, 1.6 cm) with respect to the lower left-side corner. If you move the pointer to an icon located at point (2.0 cm, 4.5 cm), what is the displacement vector of the pointer?
Strategy
The origin of the xy-coordinate system is the lower left-side corner of the computer monitor. Therefore, the unit vector
Solution
We identify
The vector component form of the displacement vector is
This solution is shown in (Figure).
Significance
Notice that the physical unit—here, 1 cm—can be placed either with each component immediately before the unit vector or globally for both components, as in (Figure). Often, the latter way is more convenient because it is simpler.
The vector x-component
Similarly, the vector y-component
The vector component form of the displacement vector (Figure) tells us that the mouse pointer has been moved on the monitor 4.0 cm to the left and 2.9 cm upward from its initial position.
Check Your Understanding A blue fly lands on a sheet of graph paper at a point located 10.0 cm to the right of its left edge and 8.0 cm above its bottom edge and walks slowly to a point located 5.0 cm from the left edge and 5.0 cm from the bottom edge. Choose the rectangular coordinate system with the origin at the lower left-side corner of the paper and find the displacement vector of the fly. Illustrate your solution by graphing.
When we know the scalar components
This equation works even if the scalar components of a vector are negative. The direction angle
When the vector lies either in the first quadrant or in the fourth quadrant, where component
Magnitude and Direction of the Displacement VectorYou move a mouse pointer on the display monitor from its initial position at point (6.0 cm, 1.6 cm) to an icon located at point (2.0 cm, 4.5 cm). What are the magnitude and direction of the displacement vector of the pointer?
Strategy
In (Figure), we found the displacement vector
Solution
The magnitude of vector
The direction angle is
Vector
Check Your Understanding If the displacement vector of a blue fly walking on a sheet of graph paper is
5.83 cm,
In many applications, the magnitudes and directions of vector quantities are known and we need to find the resultant of many vectors. For example, imagine 400 cars moving on the Golden Gate Bridge in San Francisco in a strong wind. Each car gives the bridge a different push in various directions and we would like to know how big the resultant push can possibly be. We have already gained some experience with the geometric construction of vector sums, so we know the task of finding the resultant by drawing the vectors and measuring their lengths and angles may become intractable pretty quickly, leading to huge errors. Worries like this do not appear when we use analytical methods. The very first step in an analytical approach is to find vector components when the direction and magnitude of a vector are known.
Let us return to the right triangle in (Figure). The quotient of the adjacent side
When calculating vector components with (Figure), care must be taken with the angle. The direction angle
Components of Displacement Vectors
A rescue party for a missing child follows a search dog named Trooper. Trooper wanders a lot and makes many trial sniffs along many different paths. Trooper eventually finds the child and the story has a happy ending, but his displacements on various legs seem to be truly convoluted. On one of the legs he walks 200.0 m southeast, then he runs north some 300.0 m. On the third leg, he examines the scents carefully for 50.0 m in the direction
Strategy
Let’s adopt a rectangular coordinate system with the positive x-axis in the direction of geographic east, with the positive y-direction pointed to geographic north. Explicitly, the unit vector
Solution
On the first leg, the displacement magnitude is
The displacement vector of the first leg is
On the second leg of Trooper’s wanderings, the magnitude of the displacement is
On the third leg, the displacement magnitude is
On the fourth leg of the excursion, the displacement magnitude is
On the last leg, the magnitude is
Check Your Understanding If Trooper runs 20 m west before taking a rest, what is his displacement vector?
Polar Coordinates
To describe locations of points or vectors in a plane, we need two orthogonal directions. In the Cartesian coordinate system these directions are given by unit vectors
In the polar coordinate system, the location of point P in a plane is given by two polar coordinates ((Figure)). The first polar coordinate is the radial coordinate r, which is the distance of point P from the origin. The second polar coordinate is an angle
Polar Coordinates
A treasure hunter finds one silver coin at a location 20.0 m away from a dry well in the direction
Strategy
The well marks the origin of the coordinate system and east is the +x-direction. We identify radial distances from the locations to the origin, which are
Solution
The angular coordinate of the silver coin is
For the silver coin, the coordinates are
Vectors in Three Dimensions
To specify the location of a point in space, we need three coordinates (x, y, z), where coordinates x and y specify locations in a plane, and coordinate z gives a vertical position above or below the plane. Three-dimensional space has three orthogonal directions, so we need not two but three unit vectors to define a three-dimensional coordinate system. In the Cartesian coordinate system, the first two unit vectors are the unit vector of the x-axis
In three-dimensional space, vector
If we know the coordinates of its origin
Magnitude A is obtained by generalizing (Figure) to three dimensions:
This expression for the vector magnitude comes from applying the Pythagorean theorem twice. As seen in (Figure), the diagonal in the xy-plane has length
Takeoff of a Drone
During a takeoff of IAI Heron ((Figure)), its position with respect to a control tower is 100 m above the ground, 300 m to the east, and 200 m to the north. One minute later, its position is 250 m above the ground, 1200 m to the east, and 2100 m to the north. What is the drone’s displacement vector with respect to the control tower? What is the magnitude of its displacement vector?
Strategy
We take the origin of the Cartesian coordinate system as the control tower. The direction of the +x-axis is given by unit vector
Solution
We identify b(300.0 m, 200.0 m, 100.0 m) and e(480.0 m, 370.0 m, 250.0m), and use (Figure) and (Figure) to find the scalar components of the drone’s displacement vector:
We substitute these components into (Figure) to find the displacement vector:
We substitute into (Figure) to find the magnitude of the displacement:
Check Your Understanding If the average velocity vector of the drone in the displacement in (Figure) is
35.1 m/s = 126.4 km/h
Summary
- Vectors are described in terms of their components in a coordinate system. In two dimensions (in a plane), vectors have two components. In three dimensions (in space), vectors have three components.
- A vector component of a vector is its part in an axis direction. The vector component is the product of the unit vector of an axis with its scalar component along this axis. A vector is the resultant of its vector components.
- Scalar components of a vector are differences of coordinates, where coordinates of the origin are subtracted from end point coordinates of a vector. In a rectangular system, the magnitude of a vector is the square root of the sum of the squares of its components.
- In a plane, the direction of a vector is given by an angle the vector has with the positive x-axis. This direction angle is measured counterclockwise. The scalar x-component of a vector can be expressed as the product of its magnitude with the cosine of its direction angle, and the scalar y-component can be expressed as the product of its magnitude with the sine of its direction angle.
- In a plane, there are two equivalent coordinate systems. The Cartesian coordinate system is defined by unit vectors
and along the x-axis and the y-axis, respectively. The polar coordinate system is defined by the radial unit vector , which gives the direction from the origin, and a unit vector , which is perpendicular (orthogonal) to the radial direction.
Conceptual Questions
Give an example of a nonzero vector that has a component of zero.
a unit vector of the x-axis
Explain why a vector cannot have a component greater than its own magnitude.
If two vectors are equal, what can you say about their components?
They are equal.
If vectors
If one of the two components of a vector is not zero, can the magnitude of the other vector component of this vector be zero?
yes
If two vectors have the same magnitude, do their components have to be the same?
Problems
Assuming the +x-axis is horizontal and points to the right, resolve the vectors given in the following figure to their scalar components and express them in vector component form.
a.
Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point? What is your displacement vector? What is the direction of your displacement? Assume the +x-axis is horizontal to the right.
You drive 7.50 km in a straight line in a direction
a. 1.94 km, 7.24 km; b. proof
A sledge is being pulled by two horses on a flat terrain. The net force on the sledge can be expressed in the Cartesian coordinate system as vector
A trapper walks a 5.0-km straight-line distance from her cabin to the lake, as shown in the following figure. Determine the east and north components of her displacement vector. How many more kilometers would she have to walk if she walked along the component displacements? What is her displacement vector?
3.8 km east, 3.2 km north, 2.0 km,
The polar coordinates of a point are
Two points in a plane have polar coordinates
A chameleon is resting quietly on a lanai screen, waiting for an insect to come by. Assume the origin of a Cartesian coordinate system at the lower left-hand corner of the screen and the horizontal direction to the right as the +x-direction. If its coordinates are (2.000 m, 1.000 m), (a) how far is it from the corner of the screen? (b) What is its location in polar coordinates?
Two points in the Cartesian plane are A(2.00 m, −4.00 m) and B(−3.00 m, 3.00 m). Find the distance between them and their polar coordinates.
8.60 m,
A fly enters through an open window and zooms around the room. In a Cartesian coordinate system with three axes along three edges of the room, the fly changes its position from point b(4.0 m, 1.5 m, 2.5 m) to point e(1.0 m, 4.5 m, 0.5 m). Find the scalar components of the fly’s displacement vector and express its displacement vector in vector component form. What is its magnitude?
Glossary
- component form of a vector
- a vector written as the vector sum of its components in terms of unit vectors
- direction angle
- in a plane, an angle between the positive direction of the x-axis and the vector, measured counterclockwise from the axis to the vector
- polar coordinate system
- an orthogonal coordinate system where location in a plane is given by polar coordinates
- polar coordinates
- a radial coordinate and an angle
- radial coordinate
- distance to the origin in a polar coordinate system
- scalar component
- a number that multiplies a unit vector in a vector component of a vector
- unit vectors of the axes
- unit vectors that define orthogonal directions in a plane or in space
- vector components
- orthogonal components of a vector; a vector is the vector sum of its vector components.