SAT Physics Vectors - Vector Mathematics

SAT Physics Vectors - Vector Mathematics

Components
Vectors aligned to the x- and y-axes are mathematically advantageous. However, some problems involve diagonal vector quantities. Diagonal vectors act simultaneously in both the x- and y-directions, and they are difficult to manipulate mathematically. Fortunately, diagonal vectors can be resolved into x- and y-component vectors. The x- and y-component vectors form the adjacent and opposite sides of a right triangle where the diagonal vector is its hypotenuse. Aligning the component vectors along the x- and y-axes simplifies vector addition.
The magnitudes of component vectors are determined using right triangle trigonometry. In Figure 2.5, vector A is a diagonal vector. It has a magnitude of A and a direction of θ.

The SAT Subject Test in Physics does not allow the use of a calculator. If component vectors are needed to solve a problem, there has to be an easy way to avoid using trigonometry. This can be accomplished if questions involve only three well-known, memorized, right triangles.

Important Right Triangles

On exams excluding calculator use, determining component vectors will be restricted to the key right triangles shown in Table 2.3:


A projectile is launched with an initial velocity of 50 meters per second at an angle of 37° above the horizontal. Determine the x- and y-component vectors of the velocity.

WHAT'S THE TRICK?

Draw the component vectors, and identify the adjacent and opposite sides.

The 37° angle indicates a 3-4-5 triangle. Determining the magnitudes of each component requires multiplying the hypotenuse by the correct fraction. The direction of each component can be determined by looking at the diagram.

v_x = 4/5 hyp = 4/5 (50) = 40 m/s , +x-direction
v_x = 3/5 hyp = 3/5 (50) = 30 m/s , +y-direction

In some problems, the component vectors are known or given and you must determine the vector they describe. Pythagorean theorem and inverse tangent are used to calculate the magnitude and direction of the diagonal vector described by the component vectors.

Taking an exam without using a calculator will limit your ability to perform these calculations. Problems will be limited to those easily solved by the Pythagorean theorem, or they will involve the three key right triangles described on page 64.

Adding Vectors

One important aspect of working with vectors is the ability to add two or more vectors together. Only vectors with the same units for magnitude can be added to each other. The result of adding vectors together is known as the vector sum, or resultant.
You can use two visual methods to add vectors. The first is the tip-to-tail method, and the second is the parallelogram method. In some problems, the resultant is known or given and you must determine the magnitude and direction of one of the vectors contributing to the vector sum. The sections below detail examples of each of these scenarios.

TIP-TO-TAIL METHOD

Adding vectors tip to tail is advantageous when a vector diagram is not given. Begin by sketching a coordinate axis. Vectors can be added in any order. However, drawing x-direction vectors first, followed by y-direction vectors, is advantageous. Choose the first vector, and draw it starting from the origin and pointing in the correct direction. Start drawing the tail of the next vector at the tip of the previous vector. Keep the orientation of the second vector the same as it was given in the question. Continue this process, adding any remaining vectors to the tip of each subsequent vector. Finally, draw the resultant vector from the origin (tail of the first vector) pointing to the tip of the last vector. Vector addition on the SAT Subject Test in Physics will most likely be limited to the following simple cases:

• Vectors pointing in the same direction
• Vectors pointing in opposite directions
• Vectors that are 90° apart

The following examples demonstrate tip-to-tail vector addition for these three common scenarios.

Adding Vectors Pointing in the Same Direction

A person walks 40 meters in the positive x-direction, pauses, and then walks an additional 30 meters in the positive x-direction. Determine the magnitude and direction of the person’s displacement.

WHAT'S THE TRICK?

When vectors point in the same direction, simply add them together. Sketch or visualize the vectors tip to tail. The resultant is equal to the total length of both vectors added together.

Adding Vectors Pointing in the Opposite Direction

 A person walks 40 meters in the positive x-direction, pauses, and then walks an additional 30 meters in the negative x-direction. Determine the magnitude and direction of the person’s displacement.

WHAT'S THE TRICK?

When a vector points in the opposite (negative) direction, you can insert a minus sign in front of the magnitude. Technically, vectors cannot have negative magnitudes. The minus sign actually indicates the vector’s direction, and it represents a vector turned around 180°. Again, sketching or visualizing the vectors tip to tail will help you arrive at the correct resultant. The resultant is drawn from the origin to the tip of the last vector added.

Adding Vectors That are 90° Apart

An object moves 100 meters in the positive x-direction and then moves 100 meters in the positive y-direction. Determine the magnitude and direction of the object’s displacement.

WHAT'S THE TRICK?

Start at the origin and draw the x-direction vector first. Then add the tail of the y-direction vector to the tip of the first vector. Finally, draw the resultant from the origin pointing toward the tip of the final vector added.


PARALLELOGRAM METHOD

In some exam questions, a vector diagram may be provided that shows the vectors in a tail- to-tail configuration. You can add these vectors by constructing a parallelogram, as shown in the example below.

Adding Vectors Using the Parallelogram Method

WHAT'S THE TRICK?



Finding a Missing Vector

In some problems, the resultant is known and the problem requires you to find the magnitude and direction of a missing vector. This frequently occurs when clues in the problem lead you to the conclusion that the resultant vector has a magnitude equal to zero. In order for two vectors to add up to zero, the vectors must have equal magnitudes and point in opposite directions.

Deducing the Existence of a Missing Vector

A mass, m, is initially at rest on a horizontal surface. A 10-newton force acting in the positive x-direction is applied to mass m. The mass remains at rest. Why?

WHAT'S THE TRICK?

A force is either a push or a pull. When an object remains stationary, all the pushing forces acting on the object must cancel out each other. Therefore, the sum of all the force vectors is zero. You must conclude that a second force is acting on the mass to cancel the force given in the problem. The only force capable of canceling the given force is a 10-newton force acting in the opposite direction.