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Chapter Review

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In chapter two you studied the one-dimensional forms of several quantities, such as displacement and velocity, that are associated with directions. In this chapter you learn how to extend that study to two dimensions.

3-1 Scalars versus Vectors

For many quantities used in physics, such as mass and speed, a simple number together with its units suffice to specify the quantity. Such quantities are called scalar quantities. For vector quantities the numerical value is called its magnitude. A two-dimensional vector can be represented graphically by an arrow in a coordinate system.

The vector quantity is represented in boldface, the V in the above figure. Its direction is specified by its orientation with respect to the axes of the coordinate system. In the above figure, the x-axis is used. Its magnitude is represented by the length of the arrow.

Components of a Vector

Working with vector quantities can often be simplified by resolving them into components. In two dimensions vectors have two components, one corresponding to its extent along the x-axis and the other corresponding to its extent along the y-axis. In the figure below the vector V is resolved into two scalar components V_x and V_y.

Notice that the magnitude of the vector V (not in boldface) and the two components form a right triangle. Hence, we can use the trigonometric functions to relate all of the relevant quantities

As you can see from these relations, the components can be positive, negative, or zero.

Be careful to note that the third equation above, for the angle, really only provides a reference angle for the vector with respect to one of its axes. To know precisely what the direction is, you must also account for the signs of the components, or equivalently, the quandrant of the coordinate system in which the vector lies. This last point will be illustrated in solved examples. If you know the components and want the magnitude, then the Pythagorean theorem can be used

Physlet Illustration: Vectors and Components
Start	Interactive Help on      off
A blue vector is shown on the coordinate grid.
Hints What is the x component of the vector? What is the y component of the vector? What is the magnitude of the vector? What angle does the vector make with the x axis?

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Example 3.1 Specifying a Location: You're trying to find State Park but you're lost. You ask a kind stranger for directions and he tells you that you can get there by first traveling 250 m west then 310 m north. If you wish to use a vector R to specify this location (a) what are the components of this vector? (b) what is its magnitude? and (c) specify its direction.

Picture the Problem The picture shows a coordinate system with the various directions labeled. The location is indicated by the open circle and the arrow is the vector we wish to use to specify this location. The angle q will be used to specify the direction. Try to draw R_x and R_y on this diagram yourself.

Strategy A careful reading of the problem reveals that the components are given in geographic directions; we need to translate that to the x- and y-axes. Once the components are identified exactly, we can use our knowledge of right triangles to solve the rest of the problem.

Solution

Part (a): In our coordinate system we have north +y-axis and
west -x-axis.

1. Write the x-component:	Rx = -250 m
2. Write the y-component:	Ry = 310 m
Part (b): Use the Pythagorean theorem to get the magnitude of R:
Part (c): To determine q we can use the tangent function. Which shows that the direction of R is 39o west of north.

Insight There are several points to notice here. First, even though the problem gives only positive values, because of our coordinate system R_x must be given a negative value or we would arrive at the wrong location. Also, in determining the direction, we did not just naively apply the formula tan q = R_y/R_x. When finding direction you must pay close attention to where the vector lies in the coordinate system. Because of this, I only used the absolute value of R_x so I would get a positive result for the angle.

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Physlet Illustration: Vector Components
A red vector is shown on the coordinate grid. Drag the tip of the vector until its x component is 12 units and its y component is 18 units. Start
Hints The vector's components are given in the data table. How do the numbers that appear in the table correlate with the coordinate grid that you see on the screen? Now, can you make a vector with any set of components you want? Experiment.

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Example 3.2 Designing a Garden: When planting your garden you notice that one plant is 0.75 m away from another at approximately 60 degrees above the horizontal. If you want to reproduce this pattern with another pair of plants, how far should you measure horizontally and vertically to determine where to dig the hole?

Picture the Problem The picture shows the garden and two points that mark the locations of the plants. The vector R is the position vector of one plant relative to the other. The dashed lines represent its horizontal and vertical components.

Strategy Comparing the information given with the picture, we see that the problem gives us the magnitude and direction of the position vector. Therefore, the distances we seek correspond to horizontal and vertical components of this vector.

Solution

Calculate the horizontal distance as the horizontal component of the vector R:
Calculate the corresponding vertical distance:

Insight In this example, unlike example 3.1, we started out knowing the magnitude and direction of the vector and used these to determine its components.

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Practice Quiz

Taking North to be the +y-direction and East to be +x, what are the x- and y-components (x, y) of a vector whose magnitude is 15 m and is directed 40o south of west? (11 m, 9.6 m) (-9.6 m, 11 m) (-13 m, -11 m) (9.6 m, 13 m) (-11 m, -9.6 m)

A certain vector has a y-component of 17 in arbitrary units. If its direction is 153o counterclockwise from the +x-direction, what is its magnitude? 44 37 16 -44 40

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