Statics/Vector Math

Let's say you have a box on the ground, and the box is being pulled in two directions with a certain force. You can predict the motion of the box by finding the net force acting on the box. If each force vector (where the magnitude is the tension in the rope, and the direction is the direction that the rope is "pointing") can be measured, you can add these vectors to get the net force. There are two methods for adding vectors:

Parallelogram Method

The Parallelogram Method

This is a graphical method for adding vectors. First, a little terminology:

• The tail of a vector is where it originates.
• The head of a vector is where it goes. The head is the end with the arrowhead.

This method is most easily executed using graph paper. Establish a rectangular coordinate system, and draw the first vector to scale with the tail at the origin. Then, draw the second vector (again, to scale) with its tail coincident with the head of the first vector. Then, the properties of the sum vector are as follows:

• The length of the sum vector is the distance measured from the origin to the head of the second vector.
• The direction of the sum vector is the angle.

Example

In the image at the right, the vectors (10, 53°07'48") and (10, 36°52'12") are being added graphically. The result is (19.80, 45°00'00"). (How did I measure out those angles so precisely? I did that on purpose.)

The native vector format for the parallelogram method is the 'polar form'.

Computational Method

When you use the computational method, you must resolve each vector into its x- and y-components. Then, simply add the respective components.

Converting Polar Vectors to Rectangular Vectors

If a vector is given by (r, θ), where r is the length and θ is the direction,

• x = r cos θ
• y = r sin θ

Converting Rectangular Vectors to Polar Vectors

If a vector is given by $\left \langle x, y \right \rangle$,

• $r=\sqrt{x^{2}+y^{2}}$
• $\theta=\arctan \frac{y}{x}$

Remember that the arctan() function only returns values in the range [-π/2, π/2]; therefore, if your vector is in the second or third quadrant, you will have to add π to whatever angle is returned from the arctan() function.

Example

Again referring to the image at the right, notice that the first vector can be expressed as $\left \langle 6,8 \right \rangle$, and the second is equivalent to $\left \langle 8,6 \right \rangle$. (Verify this.) Then, you simply add the components:

$\left \langle 6,8 \right \rangle + \left \langle 8,6 \right \rangle = \left \langle 6+8,8+6 \right \rangle=\left \langle 14,14 \right \rangle$


You should verify that $\left \langle 14, 14 \right \rangle$ is equal to (19.80, 45°00'00").

Multiplying Vectors

There are two ways to multiply vectors. I will not get into specific applications here; you will see many of those as you progress through the book.

The Dot Product

The dot product of two vectors results in a scalar. The dot product is the sum of the product of the components. For example:

   < 1 , 2 >
∙ < 3 , 4 >
-----------
|   +-----> 2 x 4 = 8
+---------> 1 x 3 = 3
------
11


A useful relation between vectors, their lengths, and the angle between them is given by the definition of the dot product:

$\vec a \bullet \vec b = ab \cos \theta$

• $\vec a$ and $\vec b$ are the vectors.
• $a$ and $b$ are the vectors' magnitude.
• $\theta$ is the angle between the vectors.

The Cross Product

The cross product of two vectors results in another vector. The cross product is only applicable to 3-space vectors. Remember the three unit vectors:

• $\hat i$ is the unit vector along the x-axis
• $\hat j$ is the unit vector along the y-axis
• $\hat k$ is the unit vector along the z-axis

Now if you have two vectors $\vec a = \left \langle x_1, y_1, z_1 \right \rangle$ and $\vec b = \left \langle x_2, y_2, z_2 \right \rangle$, the cross product is given by solving a determinant as follows:

$\vec a \times \vec b = \begin{vmatrix} \hat i & \hat j & \hat k \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} = \begin{vmatrix} y_1 & z_1 \\ y_2 & z_2 \end{vmatrix}\hat i + \begin{vmatrix} z_1 & x_1 \\ z_2 & x_2 \end{vmatrix} \hat j + \begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix} \hat k$


The cross product of two vectors, the lengths of those vectors, and the short angle between the vectors is given by the following relation:

$\vec a \times \vec b = ab \sin \theta$


The Right-Hand Rule

Geometrically, the cross product gives a vector that is perpendicular to the two arguments. Notice the reference to a vector, not the vector. This is because there are infinitely many vectors that are normal to two non-zero vectors. The direction of the cross product can be determined using the right-hand rule: Extend the fingers of your right hand, lay your straightened hand along the first vector, pointing your finger tips in the same direction as the vector. Curl your fingers through the short angle from the first vector to the second vector. Your thumb will point in the direction of the product vector.

Dots and Crosses of the Unit Vectors

Dot Products

• A unit vector dotted into itself gives one.
• A unit vector dotted into a different unit vector gives zero.

Cross Products

Order the unit vectors in this order: $\hat i, \hat j, \hat k$. Start at the first vector, move to the second vector, and keep going to the cross-product. If you moved immediately to the right first, the answer is positive. If you moved to the left first, the answer is negative. For example:

• $\hat i \times \hat j = \hat k$
• $\hat j \times \hat i = -\hat k$

Vector Rules

Given vectors $\vec a, \vec b, \vec c,$ and scalar r:

• $\vec a \bullet \vec b = \vec b \bullet \vec a$
• $r(\vec a \bullet \vec b) = (r \vec a) \bullet \vec b = \vec a \bullet (r\vec b)$
• $\vec a \times \vec b = -\vec b \times \vec a$
• $r(\vec a \times \vec b) = (r \vec a) \times \vec b = \vec a \times (r\vec b)$
• $(\vec a \times \vec b) \bullet \vec c = \vec a \bullet (\vec b \times \vec c)$