SAT Physics Vectors - Vectors
SAT Physics Vectors - Vectors
Although scalars possess only magnitude, vectors possess both magnitude and a specific direction.
Examples of commonly encountered vectors are listed in Table 2.2.
Formal vector variables are usually written in italics with a small arrow drawn over the letter, as shown in the middle column in Table 2.2. The College Board converted to this style in its Advanced Placement courses for the 2015 exam. Before that change, the College Board indicated vector quantities using bold standard print. Why mention this? The SAT Subject Exam in Physics is separate from the Advanced Placement Exams. As a result, there may be some differences in the conventions that these two separate College Board organizations use to express vector quantities.
format used in this book and in the SAT Subject Test in Physics. The second, F, is an alternate way to indicate a vector quantity. The next two, Fx and Fy, signify vector components that lie along the specific axis indicated by their subscripts. The last, F, appears to be the convention to indicate a scalar quantity. This is actually used very frequently in textbooks and on formal exams. It is typically used when only the magnitude of the vector is needed and the direction is understood.
Distinguishing between vectors and scalars by simply looking at an equation can be confusing. How, then, do we tell scalars and vectors apart? Physics problems may contain clues in the text of the problem to help distinguish vectors from scalars. The mention of a specific direction definitely indicates a vector quantity. However, it is up to the student to learn which quantities are vectors and when the use of vector components is necessary. Counting on the use of a specific set of symbol conventions may not be wise.
Vectors do follow certain mathematical conventions that are worth noting. Vector magnitudes can be only positive or zero. However, vectors can have negative direction. Consider the acceleration of gravity, a vector quantity acting in the negative y-direction. The gravity vector includes both magnitude and directionSubstituting this exact expression, including the negative y-direction, into an equation is not really workable. Instead the value -10 m/s2 may be substituted into equations. The negative sign in front of the magnitude indicates the negative y-direction. This can be done only if all the vector quantities used in an equation lie along the same axis and it is understood that the signs on all vector quantities represent direction along that axis. This essentially transforms the vector quantities into scalar quantities, allowing normal mathematical operations. As a result, the variable may be shown as a scalar in italics (g= -10 m/s2) rather than in bold print. When a negative sign is associated with a vector quantity, it technically specifies the vector’s direction and assists with proper vector addition.
Vectors are represented graphically as arrows. For displacement vectors, the tail of the arrow is the initial position of the object, xf and the tip of the arrow is the final position of the object, xi The length of the arrow represents the vector’s magnitude, and its orientation on the coordinate axis indicates direction. This may give some insight into the reason that some vector quantities are displayed in italics.
Figure 2.3 shows a car moving 200 meters and its associated vector.
The magnitude of the displacement vector, Δx, is the absolute value of the difference between the final position, xf and the initial position, xi. Direction can be seen in the diagram.
Δx=xf-xi=200-0 = 200 m, to the right (+x)For other vectors, such as velocity and force, the quantity described by the vector occurs at the tail of the arrow. The tail of the arrow shows the actual location of the object being acting upon by the vector quantity. The tip of the arrow points in the direction the vector is acting. The length of the arrow represents the magnitude of the vector quantity. The magnitude and direction described by these types of vectors may be instantaneous values capable of changing as the object moves. In addition, the object may not reach the location specified by the tip of the arrow.
These types of vectors are readily seen in projectile motion. In Figure 2.4, a projectile is launched with a speed of 50 meters per second at an angle of 37° above the horizontal.
Although only three key velocity vectors are shown in the diagram, they clearly demonstrate how the magnitude and direction of velocity change throughout the flight. During the motion depicted in the diagram, no two instantaneous velocity vectors are completely alike.
You will encounter a variety of vector quantities in the chapters ahead. Knowing how to recognize vectors quantities like displacement, velocity, acceleration, and force will improve your problem-solving skills. The importance of vector direction cannot be overstated. Including the correct sign representing a vector’s direction is often the key to arriving at the correct solution. The next sections will demonstrate the importance of vector direction as we review basic vector mathematics.