SAT Physics Subject Test

Chapter 1 Math Review

VECTORS

Definition

vector is a quantity that contains information about 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, 55 miles per hour is a scalar quantity, while 55 miles per hour, to the north is a vector quantity. Speed and distance are scalar quantities. Other examples of scalars include: mass, work, energy, power, temperature, and electric charge.

Distance

Distance is a scalar
quantity. It refers to the
amount of ground an
object has covered.

The scalars of distance and speed are paired with the vectors of displacement and velocity, respectively.

Vectors can be denoted in several ways, including:

In textbooks, you’ll usually see the first one, but when it’s handwritten you’ll see one of the last two. In this book we will show all vector quantities in bold. For example, A would be the scalar quantity, and A the vector quantity.

Graphically, a vector is represented as an arrow whose length represents the magnitude and whose direction represents, well, the direction.

Displacement (which is net distance [magnitude] traveled plus direction) is the prototypical example of a vector:

Displacement

Displacement is a vector
quantity. It refers to how
far out of place an object
is from its original
position.

When we say that vectors obey the commutative law of 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 following 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:


A + B = B + A


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. The vector formed is the sum of the first two. This is called the “tip-to-tail” method of vector addition.

1. Add the following two vectors.

Here’s How to Crack It

Place the tail of B at the tip of A and connect them:

Scalar Multiplication of Vectors

A vector can be multiplied by a scalar (that is, by a regular number), which results in a vector. Scalars scale a vector in the sense that they alter the magnitude but not the direction. If the original vector is A and the scalar is 4, then the scalar multiple 4A. Simply put, it has a magnitude that is four times greater than the original vector. The vector −4A would also be four times greater than A, but would point in the opposite direction.


magnitude of κA = |κ|× (magnitude of A)


2. Sketch the scalar multiples 2AA, –A, and –3A of the vector A.

Here’s How to Crack It

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. (Note that, unlike vector addition, vector subtraction is not commutative.)

A – B = A + (–B)

3. For the two vectors A and B, find the vector A – B.

Here’s How to Crack It

Flip B around—thereby forming –B—and add that vector to A.

Components of Vectors

So as you can see, a vector can be defined as the sum of two (or more) vectors. Perpendicular vectors that are added together to make up a vector are called its components. The vectors B and C, below, are called the vector components of A. 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.

B + C = A

Since we are working with two dimensions, it is usually easiest to choose component vectors that lie along the x- and y- axes of the rectangular coordinate system, also known as the Cartesian coordinate system. The three arrows of the known vector and its component vectors create a right triangle. Setting up the components of the vectors in this way makes it much easier to add and subtract vectors and it allows you to use the Pythagorean theorem instead of some tricky trig.

(Ax)2 + (Ay)2 = A

In the figure above, vector A, which is in the Cartesian plane is made up of the components Ax along the x-axis and Ay along the y-axis. Ax and Ay are called the scalar components of A.

A vector can be expressed in terms of its components using the unit vectors i and j.

i is a vector of magnitude one that points in the positive x direction, and j is a vector of magnitude one that points in the positive y direction.

If the components of A are Ax and Ay, then A = Axi + Ayj.

Vector Operations Using Components

Using perpendicular components makes the vector operations of addition, subtraction, and scalar multiplication pretty straightforward.

Vector Addition

Vectors A and B below are added together to form vector C.

C = A + B

Cx = Ax + Bx

Cy = Ay + By

To add two or more vectors, resolve each vector into its horizontal and vertical components. Add the components along the x-axis to form the x-component of the resultant vector, and then add the components along the y-axis to form the y-component of the resultant vector. You can use the values for Cx and Cy along with the Pythagorean theorem to determine the magnitude and direction of the resultant vector.

Vector Subtraction

To subtract vector B from vector A, use the same procedure. Resolve each vector into perpendicular components and subtract them in the indicated order.

C = A − B

Cx = Ax − Bx

Cy = Ay − By

Scalar Multiplication

Scalar multiplication just means that you increase the scale of the vector. (Or decrease it, if you multiply by a fraction.) Multiply each component by a given number.

3A = 3Ax + 3Ay

   4. If the components of A are Ax = 2 and Ay = –3, and the components of B are Bx = –4 and By = 2, compute the components of each of the following vectors.

(A) A + B

(B) A − B

(C) 2A

(D) A + 3B

Here’s How to Crack It

(A)   Using unit vector notation, A = 2i – 3j and B = – 4i +2j. Adding components, we see that A + B = (2 + [–4])i + ([–3] + 2)j = – 2i – j. Therefore, the x-component of the sum is – 2 and the y-component is – 1.

(B)   A – B = (2i – 3j) – (– 4i + 2j) = (2 – [–4])i + (−3 – 2)j = 6i – 5j. The x-component is 6 and the y-component is –5.

(C)   2A = 2(2i – 3j) = 4i – 6j. 4 and – 6 are the x- and y-components, respectively.

(D)   A + 3B = (2i – 3j) + 3(–4i + 2j) = (2 + 3 [–4])i + (–3 + 3 [2])j = –10i + 3j. –10 and 3 are the x- and y-components, respectively.

Magnitude of a Vector

Magnitude is a scalar number indicating the length of a vector. Use the Pythagorean theorem! You can use components Cx and Cy to find the magnitude of the new vector C.

In this example, Cx = 13, Cy = 18, and vector C is the hypotenuse of the two.

Magnitude of a Vector Using SOHCAHTOA

You can use SOHCAHTOA to find Ax and Ay values. If A makes the angle θ with the x-axis, then its x- and y-components are A cosθ and A sinθ, 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 cosθ and W sinθ:

Summary

·        Use SOHCAHTOA to remember the definitions of sinθ, cosθ, and tanθ.

·        sinθ = 

·        cosθ = 

·        tanθ = 

·        A vector is a quantity that has a direction as well as a magnitude.

·        Add vectors by connecting vector arrows tip to tail and connecting the exposed tail to the exposed tip. The vector formed is the sum of the other vectors.

·        Because A − B = A + (−B), you can subtract a vector by multiplying it by −1 (flipping the point of the arrow to the opposite end) and adding the resultant vector to the other.

·        You can also add or subtract vectors by adding or subtracting their components.

·        Multiplying a vector by a positive scalar creates a vector in the same direction. Multiplying a vector by a negative scalar creates a vector in the opposite direction.

·        To find the magnitude (length) of a vector, you cannot simply add the lengths of the other two vectors. Resolve the vectors into horizontal and vertical components and use the Pythagorean theorem.