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Vectors

A vector is a quantity with a magnitude and direction. In these notes, a vector quantity will be denoted by a bold letter (such as the electric field vector \(\mathbf{E}\)). Graphically, vectors are represented as straight arrows.  The length of the arrow typically represents the vector's magnitude, and the arrow points in the same direction as the vector.

Adding vectors

It is often useful to combine vectors by vector addition. For example, if we are looking for the total momentum of a system we add all the momentum vectors. To find the net force on an object, we add the vectors of all forces acting on the object.  To find the total electric field at a particular location, we add all the vectors of all electric fields at that location together. While these examples all refer to different physical situations, the way we add vectors together is the same.  However, vectors do not add like numbers!

Graphical Addition

The grid below shows two vectors \(\mathbf{A}\) and \(\mathbf{B}\).

To add these vectors we join the arrows up to make a “path” which we can follow, always going in the direction of the arrows. The vector \(\mathbf{A + B}\) is the vector that connects the beginning of this path to the end, as shown below.

Component Addition

Another method for adding vectors is by breaking a vector up into components.  While vectors don’t add like numbers, the components of a vector do. We will break this vector into \(x\)- and \(y\)-components, which are the most common choice. To do this we must figure out how many units the vector points right (\(+x\)) and how many units the vectors points up (\(+y\)). Sometimes a grid is used to assist in this, like in the example above.  In this case, we can just count the number of units, but in many situations we will have to use trigonometry to break a vector into components. From the example above,

\[\textbf{A} = 6\textrm { units right, } -3 \textrm{ units up}\]

\[\textbf{B} = 4 \textrm{ units right, } 4 \textrm{ units up}\]

\[\textbf{A + B}= 10\textrm{ units right, } \mathbf{1 } \textrm{ unit up}\]

Subtraction

 To subtract \(\mathbf{B}\) from \(\mathbf{A}\), we add vectors \(\mathbf{A}\) and \(\mathbf{-B}\).  \(\mathbf{-B}\) is the negative of \(\mathbf{B}\), defined as \(- \mathbf{B} \equiv (-1)\mathbf{B}\).  Graphically, \(- \mathbf{B}\) is an arrow with the same magnitude as \(\mathbf{B}\), but pointing in the opposite direction.

\[\mathbf{A - B} = \mathbf{A} + (\mathbf{-B})\]

Applying this method to the vector components is a valid way to subtract vectors mathematically.  This gives us

\[\textbf{A - B}= (6-4)  \textrm{ units right, } (-3-4) \textrm{ units up}\]

\[\textbf{A - B} =2  \textrm{ units right, } -7 \textrm{ units up}\]

Magnitude 

The magnitudes of \(\mathbf{A}, \mathbf{B},\) and \(\mathbf{A + B}\) are denoted \(|\mathbf{A}|, |\mathbf{B}|,\) and \(|\mathbf{A + B}|\), respectively.  Using the Pythagorean Theorem, we can show:

\[|\mathbf{A}| = \sqrt{(6)^2 + (-3)^2} \textrm{ units} = 6.71 \textrm{ units}\]

\[|\mathbf{B}| = \sqrt{(4)^2 + (4)^2} \text{ units} = 5.66 \textrm{ units}\]

\[|\mathbf{A + B}| = \sqrt{(6+4)^2 + (-3+4)^2} \text{ units} = 10.05 \textrm{ units}\]

It's clear to see in this case how the magnitudes of vectors do not add like numbers, that is to say \(|\mathbf{A}| + |\mathbf{B}| \neq |\mathbf{A + B}|\).   The same is true for subtraction \(|\mathbf{A}| - |\mathbf{B}| \neq |\mathbf{A - B}|\).

Scalar Multiplication

Multiplying a vector by a positive number changes the magnitude of the vector, but does not change the direction.  Multiplying \(\mathbf{A}\) by 2 gives us \(\mathbf{2A}\), which points in the same direction as \(\mathbf{A}\) but is twice as long.

Multiplying a vector by a negative number changes the magnitude of the vector and makes it point in the opposite direction.  The vector \(\mathbf{-2A}\) is twice as long as \(\mathbf{A}\) and points in the opposite direction.

If we multiply a vector by a number with units, the final vector has a magnitude with these new units as well.  Consider this equation for the Lorentz force.

\[\mathbf{F} = q\mathbf{E}\]

  • The units of \(\mathbf{F}\) are \(\text{N}\).  The units of \(q\) are \(\text{C}\).  The units of \(\mathbf{E}\) are \(\dfrac{N}{C}\).
  • If \(q\) is positive, \(\mathbf{F}\) and \(\mathbf{E}\) have the same direction.
  • If \(q\) is negative, \(\mathbf{F}\) and \(\mathbf{E}\) have the opposite direction.

Vectors Into/Out of Page

Scientists have adopted a special notation to help with drawing vectors in three dimensions when your paper (or computer screen) has only two dimensions. Vectors that point in a direction perpendicular to the page are represented with these symbols:

  • The symbol on the left with an X represents a vector pointing into the page.
  • The symbol on the right with a dot represents a vector pointing out of the page.
Questions?

This page is a brief review of material that was covered in Physics 7B. If you still find vectors confusing, look back through your 7B notes, or talk to your TA during office hours.