Vector Algebra

 Vector Algebra

Before you start this lab, first read the general instructions for the PEP 294 web-based labs.

Procedures

Trigonometric functions are useful mathematical tools by which a vector quantity may be resolved into horizontal and vertical components, and the angle at which these components function. For instance, in the long jump event in track and field, it is helpful to know what angle is necessary at takeoff in order to produce the greatest amount of distance. By calculating the optimal angle for takeoff, it becomes easier to quantify what the horizontal and vertical velocity of the athlete must be to achieve the greatest distance. Three trigonometric functions we will utilize are the sine, cosine, and tangent function.
A vector quantity has both magnitude and direction. Due to this fact, we need a special way to perform vector algebra such as addition and subtraction. Subtraction is nothing more than a special case of addition.

Stage 1: Trigonometric Functions
Figure 1 shows a right-angled triangle in which one of its inside angles is right angle (90 degrees). Three trigonometric functions are commonly used: sine (sin), cosine (cos) and tangent (tan):

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SOH CAH TOA

[Example 1]    In Figure 1, let the opposite = 1.0, the adjacent = 1.732 and the hypotenuse = 2.0, then what are sin, cos and tan?

- Based on the definitions of the trigonometric functions:

[Example 2]    Using your calculator, compute sin 30°, cos 25° and tan 66° (3 decimal places).

- Use the 'sin', 'cos' and 'tan' functions available in your calculator:

If you, by chance, get values different from those shown above, it means your unit of angle is now set to 'radian' rather than 'degree'. Change the unit to 'degree'. Consult the your calculator's manual for this.

In many cases, we need to compute the angles from given trig. values. The inverse trigonometric functions available in your calculator are used in these cases:

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Use the '2nd Function' button in your calculator to invoke these functions.

[Example 3]   Compute angle in the following cases.

- Use the inverse trigonometric functions:

Stage 2: Vector Addition - The Graphical Method (The Tip-to-Tale Method)
A vector represents a quantity that contains both magnitude and direction. Vectors may be added or subtracted, in order to determine the net vector, otherwise known as the resultant vector. When vectors of the same nature exist in the same plane it is possible to add them by the tip-to-tail method. In this method, the tail of one vector is placed at the tip of another vector. The resultant vector is the result of all the vectors added as shown in Figure 2. This is due to the fact that the vector quantity has both direction & magnitude.
In the process of vector addition, one can freely move the vectors keeping the directions and the magnitudes unaltered. The sequence of adding vectors does not matter in this process. Click here to download a Word document file which shows this.

Stage 3: Vector Resolution and Composition
One can resolve a vector into components vectors. Figure 3 illustrates this process.
Vector a (Figure 3a) is resolved into vectors a_X and a_Y shown in Figure 3b. According to the tip-to-tail method, sum of vectors a_X and a_Y is the same to vector a. Vectors a_X and a_Y are parallel to the X and Y axis, respectively, and called as the component vectors. The component vectors are perpendicular to each other. Vector a and the two component vectors form a right-angled triangle:

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where, = the angle of the vector and the horizontal axis. Use Equation 3 to compute the magnitudes of the component vectors. Pay attention to the direction of the component vectors. By convention, rightward and upward are the positive directions while the leftward and downward directions are negative.

[Example 4]     A soccer ball is kicked at an angle of 35° from the ground (Figure 4). If the velocity of the ball is 20 m/s, compute the horizontal and vertical velocities of the ball at the instant of kick.

- Velocity is a vector quantity, so it can be resolved into components: horizontal & vertical. Using Equation 3:

The steps to follow in this process are:

1. Draw a right-angled triangle using the vector and the axes (click here to see an example).
2. Identify the hypotenuse of the right-angled triangle.
3. Compute the magnitudes of the component vectors.
4. Check the directions of the component vectors and add '-' for the leftward (westward) and downward (southward) components.

When the component vectors are known, one can compose the actual vector using the component vectors. This process is called vector composition. Again, component vectors a_X and a_Y, and the resultant vector (a) in Figure 3b form a right-angled triangle. From the Pythagorean Theorem:

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[Example 5]     A swimmer orients herself perpendicular to the parallel banks of a river (Figure 5). If the swimmer's velocity is 2 m/s and the velocity of the current is 0.5 m/s, what will be the swimmer's resultant velocity?  (Hall, S.J. (1995). Basic Biomechanics, 2nd Ed. pp. 302.)

- The swimming velocity and the velocity of the current are perpendicular to each other. In other words, these two velocity vectors can be regarded as the component vectors of the resultant velocity vector (v). Applying Equation 4:

The steps need to follow to solve this type of problem are:

1. Draw the diagram (right-angled triangle) formed by the two component vectors and the resultant vector.
2. Use the Pythagorean Theorem to compute the magnitude of the resultant vector.
3. Use the inverse tangent function to compute the direction of the resultant vector.

Stage 4: Vector Addition - The Component Method
One can compute the resultant vector (sum vector) of vectors of the same nature through the component approach. Let C be the sum of vectors A and B shown in Figure 6a. C can be visualized through the tip-to-tale method (Figure 6b). As shown in Figures 6b and 6c, the following relationships are holding among the components of the vectors:

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In other words, the components of the resultant vector (C) can be directly computed from the components of the individual vectors (A and B) through addition, component by component. Once the components of vector C are known, the magnitude (C) and the direction (q) can be computed.
The key to this approach is the combination of vector resolution and composition. Here are the general steps:

1. Add the vectors graphically using the tip-to-tale method and visualize the resultant vector.
2. Compute the components of the individual vectors to be added. Pay attention to the direction of the components. Add '-' to the leftward or downward components.
3. Compute the components of the resultant vector by adding the vectors component by component.
4. Compute the magnitude of the resultant vector using the Pythagorean Theorem.
5. Compute the direction of the resultant vector using an inverse trig function.

[Example 6]  An orienteer runs 400 m directly east and then 500 m to northeast (at a 45° angle from due east and from due north). Compute the overall displacement. (Hall, S.J. (1995). Basic Biomechanics, 2nd Ed. pp. 328.)

- Components of the individual vectors:

- Compute the components of the resultant vector:

- Compute the magnitude and direction of the resultant vector:

- Therefore, the resultant displacement is 832.30 m and its direction is 25.13° N of due E.

[Example 7]  The same orienteer this time runs 400 m directly eastward, then 500 m to 60° N of due W. Compute the overall displacement.

- Draw the resultant vector using the tip-to-tale method:

- Compute the components of the individual vectors:

- Compute the components of the resultant vector:

- Compute the magnitude and the direction of the resultant vector:

- The resultant displacement is 458.25 m and its direction is 70.89° N of due E.

Summary

Trigonometric functions are commonly used in the analysis of human motion in conjunction with vectors. A vector has both magnitude and direction. Due to the direction information, it is impossible to directly add or subtract vectors. Many biomechanical quantities such as position, displacement, velocity, acceleration, momentum, and force are vectors. Therefore, it is very important for the students to understand how to manipulate vectors and how to utilize the trigonometric functions.

Questions (20 pts)

1. For the right-angled triangle shown in Figure 1 above, opposite = 3, and adjacent = 5.

(a) Compute the hypotenuse (2 decimal places).
(b) Compute sin (3 decimal places).
(c) Compute (2 decimal places).

2. A sailboat is being propelled in the direction of 40° W of due N at a speed of 3 m/s.

(a) Draw the diagram showing the situation.
(b) Resolve the velocity vector into components. Compute the velocity components and describe their directions.

3. A buoy is floating on the water. The current carries the buoy southward at a velocity of 0.75 m/s while the wind blows the buoy westward at 0.60 m/s.

(a) Draw the diagram which shows the velocity components and the resultant velocity vector. Pay attention to the direction of the resultant velocity vector.
(b) Compute the magnitude of the resultant velocity.
(c) Compute the direction of the resultant velocity.

4. An orienteer runs 500 m in the direction of 45° N of due E and then 200 m in the direction of 30° W of due N. (Click here to see the diagram.)

(a) Compute the components of the two displacement vectors.
(b) Compute the components of the resultant displacement vector.
(c) Compute the magnitude and the direction of the resultant vector.