AP Physics B Exam
Vectors will show up all over the place in our study of physics. Some physical quantities that are represented as vectors are displacement, velocity, acceleration, force, momentum, and electric and magnetic fields. Since vectors play such a recurring role, it’s important to become comfortable working with them; the purpose of this chapter is to provide you with a mastery of the fundamental vector algebra we’ll use in subsequent chapters. For now, we’ll restrict our study to two-dimensional vectors (that is, ones that lie flat in a plane).
A vector is a quantity that involves both magnitude and direction and obeys the commutative law for addition, which we’ll explain in a moment. A quantity that does not involve direction is a scalar. For example, the quantity 55 miles per hour is a scalar, while the quantity 55 miles per hour to the north is a vector. Other examples of scalars include: mass, work, energy, power, temperature, and electric charge.
Vectors can be denoted in several ways, including:
A, A, , Ā
In textbooks, you’ll usually see one of the first two, but when it’s handwritten, you’ll see one of the last two.
Displacement (which is net distance traveled plus direction) is the prototypical example of a vector:
When we say that vectors obey the commutative law for addition, we mean that if we have two vectors of the same type, for example, another displacement,
then A + B must equal B + A. The vector sum A + B means the vector A followed by B, while the vector sum B + A means the vector B followed by A. That these two sums are indeed identical is shown in the following figure:
Two vectors are equal if they have the same magnitude and the same direction.
VECTOR ADDITION (GEOMETRIC)
The figure above illustrates how vectors are added to each other geometrically. Place the tail (the initial point) of one vector at the tip of the other vector, then connect the exposed tail to the exposed tip. It is essential that the original magnitude and direction of each vector be preserved. The vector formed is the sum of the first two. This is called the “tip-to-tail” method of vector addition.
Example 1.1 Add the following two vectors:
Solution. Place the tail of B at the tip of A and connect them:
A vector can be multiplied by a scalar (that is, by a number), and the result is a vector. If the original vector is A and the scalar is k, then the scalar multiple kA is as follows:
magnitude of kA = |k × (magnitude of A)
Example 1.2 Sketch the scalar multiples 2A, A, –A, and –3A of the vector A:
VECTOR SUBTRACTION (GEOMETRIC)
To subtract one vector from another, for example, to get A – B, simply form the vector –B, which is the scalar multiple (–1)B, and add it to A:
A – B = A + (–B)
Example 1.3 For the two vectors A and B, find the vector A – B.
Solution. Flip B around—thereby forming –B—and add that vector to A:
STANDARD BASIS VECTORS
Two-dimensional vectors, that is, vectors that lie flat in a plane, can be written as the sum of a horizontal vector and a vertical vector. For example, in the following diagram, the vector A is equal to the horizontal vector B plus the vertical vector C:
The horizontal vector is always considered a scalar multiple of what’s called the horizontal basis vector, i, and the vertical vector is a scalar multiple of the vertical basis vector, j. Both of these special vectors have a magnitude of 1, and for this reason, they’re called unit vectors. Unit vectors are often represented by placing a hat (caret) over the vector; for example, the unit vectors i and j are sometimes denoted and .
For instance, the vector A in the figure below is the sum of the horizontal vector B = 3 and the vertical vector C = 4.
The vectors B and C are called the vector components of A, and the scalar multiples of and which give A—in this case, 3 and 4—are called the scalar components of A. So vector A can be written as the sum Ax + Ay, where Ax and Ay are the scalar components of A. The componentAx is called the horizontal scalar component of A, and Ay is called the vertical scalar component of A. In general, any vector in a plane can be described in this manner.
VECTOR OPERATIONS USING COMPONENTS
The use of components makes the vector operations of addition, subtraction, and scalar multiplication pretty straightforward:
Vector addition: Add the respective components.
A + B = (Ax + Bx) + (Ay + By)
Vector subtraction: Subtract the respective components.
A – B = (Ax – Bx) + (Ay – By)
Scalar multiplication: Multiply each component by k.
kA = (kAx) + (kAy)
Example 1.4 If A = 2 – 3 and B = –4 + 2, compute each of the following vectors: A + B, A – B, 2A, and A + 3B.
Solution. It’s very helpful that the given vectors A and B are written explicitly in terms of the standard basis vectors and :
A + B = (2 – 4) + (–3 + 2) = –2 –
A – B = [2 – (–4)] + (–3 –2) = 6 – 5
2A= 2(2) + 2(–3) = 4 – 6
A + 3B = [2 + 3(–4)] + [–3 + 3(2)] = –10 + 3
MAGNITUDE OF A VECTOR
The magnitude of a vector can be computed with the Pythagorean theorem. The magnitude of vector A can be denoted in several ways: A or |A| or ||A||. In terms of its components, the magnitude of A = Ax + Ay is given by the equation
which is the formula for the length of the hypotenuse of a right triangle with sides of lengths Ax and Ay.
DIRECTION OF A VECTOR
The direction of a vector can be specified by the angle it makes with the positive x axis. You can sketch the vector and use its components (and an inverse trig function) to determine the angle. For example, if θ denotes the angle that the vector A = 3 + 4 makes with the +x axis, then tan θ = 4/3, so θ = tan–1(4/3) = 53.1°.
In general, the axis that θ is made to is known as the adjacent axis. The adjacent component is always going to get the cos θ. For example, if A makes the angle θ with the +x axis, then its x- and y-components are A cos θ and A sin θ, respectively (where A is the magnitude of A).
In general, any vector in the plane can be written in terms of two perpendicular component vectors. For example, vector W (shown below) is the sum of two component vectors whose magnitudes are W cos θ and W sin θ: