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Section 5-4 : Cross Product
In this final section of this chapter we will look at the cross product of two vectors. We should note that the cross product requires both of the vectors to be three dimensional vectors.
Also, before getting into how to compute these we should point out a major difference between dot products and cross products. The result of a dot product is a number and the result of a cross product is a vector! Be careful not to confuse the two.
So, let’s start with the two vectors (vec a = leftlangle {{a_1},{a_2},{a_3}} rightrangle ) and (vec b = leftlangle {{b_1},{b_2},{b_3}} rightrangle ) then the cross product is given by the formula,
[vec a times vec b = leftlangle {{a_2}{b_3} - {a_3}{b_2},{a_3}{b_1} - {a_1}{b_3},{a_1}{b_2} - {a_2}{b_1}} rightrangle ]
This is not an easy formula to remember. There are two ways to derive this formula. Both of them use the fact that the cross product is really the determinant of a 3x3 matrix. If you don’t know what that is don’t worry about it. You don’t need to know anything about matrices or determinants to use either of the methods. The notation for the determinant is as follows,
[vec a times vec b = left| {begin{array}{*{20}{c}}{vec i}&{vec j}&{vec k}{{a_1}}&{{a_2}}&{{a_3}}{{b_1}}&{{b_2}}&{{b_3}}end{array}} right|]
The first row is the standard basis vectors and must appear in the order given here. The second row is the components of (vec a) and the third row is the components of (vec b). Now, let’s take a look at the different methods for getting the formula.
The first method uses the Method of Cofactors. If you don’t know the method of cofactors that is fine, the result is all that we need. Here is the formula.
[vec a times vec b = left| {begin{array}{*{20}{c}}{{a_2}}&{{a_3}}{{b_2}}&{{b_3}}end{array}} right|vec i - left| {begin{array}{*{20}{c}}{{a_1}}&{{a_3}}{{b_1}}&{{b_3}}end{array}} right|vec j + left| {begin{array}{*{20}{c}}{{a_1}}&{{a_2}}{{b_1}}&{{b_2}}end{array}} right|vec k]
where,
[left| {begin{array}{*{20}{c}}a&bc&dend{array}} right| = ad - bc]
This formula is not as difficult to remember as it might at first appear to be. First, the terms alternate in sign and notice that the 2x2 is missing the column below the standard basis vector that multiplies it as well as the row of standard basis vectors.
The second method is slightly easier; however, many textbooks don’t cover this method as it will only work on 3x3 determinants. This method says to take the determinant as listed above and then copy the first two columns onto the end as shown below.
[vec a times vec b = left| {begin{array}{*{20}{c}}{vec i}&{vec j}&{vec k}{{a_1}}&{{a_2}}&{{a_3}}{{b_1}}&{{b_2}}&{{b_3}}end{array}} right|,begin{array}{*{20}{c}}{vec i}&{vec j}{{a_1}}&{{a_2}}{{b_1}}&{{b_2}}end{array}]
We now have three diagonals that move from left to right and three diagonals that move from right to left. We multiply along each diagonal and add those that move from left to right and subtract those that move from right to left.
This is best seen in an example. We’ll also use this example to illustrate a fact about cross products.
Example 1 If (vec a = leftlangle {2,1, - 1} rightrangle ) and (vec b = leftlangle { - 3,4,1} rightrangle ) compute each of the following.
(vec a times vec b)
(vec b times vec a)
Show All SolutionsHide All Solutionsa (vec a times vec b) Show Solution
Here is the computation for this one.
[begin{align*}vec a times vec b & = left| {begin{array}{*{20}{c}}{vec i}&{vec j}&{vec k}2&1&{ - 1}{ - 3}&4&1end{array}} right|,begin{array}{*{20}{c}}{vec i}&{vec j}2&1{ - 3}&4end{array} & = vec ileft( 1 right)left( 1 right) + vec jleft( { - 1} right)left( { - 3} right) + vec kleft( 2 right)left( 4 right) - vec jleft( 2 right)left( 1 right) - vec ileft( { - 1} right)left( 4 right) - vec kleft( 1 right)left( { - 3} right) & = 5vec i + vec j + 11vec kend{align*}] b (vec b times vec a) Show Solution
Notice that switching the order of the vectors in the cross product simply changed all the signs in the result. Note as well that this means that the two cross products will point in exactly opposite directions since they only differ by a sign. We’ll formalize up this fact shortly when we list several facts.
There is also a geometric interpretation of the cross product. First we will let (theta) be the angle between the two vectors (vec a) and (vec b)and assume that (0 le theta le pi ), then we have the following fact,
[begin{equation}left| {vec a times vec b} right| = left| {vec a} right|,left| {vec b} right|,sin theta label{eq:eq1} end{equation}]
and the following figure.
There should be a natural question at this point. How did we know that the cross product pointed in the direction that we’ve given it here?
First, as this figure implies, the cross product is orthogonal to both of the original vectors. This will always be the case with one exception that we’ll get to in a second.
Second, we knew that it pointed in the upward direction (in this case) by the “right hand rule”. This says that if we take our right hand, start at (vec a) and rotate our fingers towards (vec b)our thumb will point in the direction of the cross product. Therefore, if we’d sketched in (vec b times vec a) above we would have gotten a vector in the downward direction.
Example 2 A plane is defined by any three points that are in the plane. If a plane contains the points (P = left( {1,0,0} right)), (Q = left( {1,1,1} right)) and (R = left( {2, - 1,3} right)) find a vector that is orthogonal to the plane. Show Solution
The one way that we know to get an orthogonal vector is to take a cross product. So, if we could find two vectors that we knew were in the plane and took the cross product of these two vectors we know that the cross product would be orthogonal to both the vectors. However, since both the vectors are in the plane the cross product would then also be orthogonal to the plane.
So, we need two vectors that are in the plane. This is where the points come into the problem. Since all three points lie in the plane any vector between them must also be in the plane. There are many ways to get two vectors between these points. We will use the following two,
So, the vector (4vec i + vec j - vec k) will be orthogonal to the plane containing the three points.
Now, let’s address the one time where the cross product will not be orthogonal to the original vectors. If the two vectors, (vec a) and (vec b), are parallel then the angle between them is either 0 or 180 degrees. From (eqref{eq:eq1}) this implies that,
[left| {vec a times vec b} right| = 0]
From a fact about the magnitude we saw in the first section we know that this implies
[vec a times vec b = vec 0]
In other words, it won’t be orthogonal to the original vectors since we have the zero vector. This does give us another test for parallel vectors however.
Fact
If (vec a times vec b = vec 0) then (vec a) and (vec b) will be parallel vectors.
Let’s also formalize up the fact about the cross product being orthogonal to the original vectors.
Fact
Provided (vec a times vec b ne vec 0) then (vec a times vec b) is orthogonal to both (vec a) and (vec b).
Here are some nice properties about the cross product.
Properties
If (vec u), (vec v) and (vec w) are vectors and (c) is a number then,
[begin{align*} & vec u times vec v = - vec v times vec u & hspace{0.75in} & left( {cvec u} right) times vec v = vec u times left( {cvec v} right) = cleft( {vec u times vec v} right) &vec u times left( {vec v + vec w} right) = vec u times vec v + vec u times vec w & hspace{0.75in} & vec ucenterdot left( {vec v times vec w} right) = left( {vec u times vec v} right)centerdot vec w & vec ucenterdot left( {vec v times vec w} right) = left| {begin{array}{*{20}{c}}{{u_1}}&{{u_2}}&{{u_3}}{{v_1}}&{{v_2}}&{{v_3}}{{w_1}}&{{w_2}}&{{w_3}}end{array}} right| & & end{align*}]
The determinant in the last fact is computed in the same way that the cross product is computed. We will see an example of this computation shortly.
There are a couple of geometric applications to the cross product as well. Suppose we have three vectors (vec a), (vec b) and (vec c) and we form the three dimensional figure shown below.
The area of the parallelogram (two dimensional front of this object) is given by,
[{rm{Area}} = left| {vec a times vec b} right|]
and the volume of the parallelepiped (the whole three dimensional object) is given by,
[{rm{Volume}} = left| {vec acenterdot left( {vec b times vec c} right)} right|]
Note that the absolute value bars are required since the quantity could be negative and volume isn’t negative.
We can use this volume fact to determine if three vectors lie in the same plane or not. If three vectors lie in the same plane then the volume of the parallelepiped will be zero.
Example 3 Determine if the three vectors (vec a = leftlangle {1,4, - 7} rightrangle ), (vec b = leftlangle {2, - 1,4} rightrangle ) and (vec c = leftlangle {0, - 9,18} rightrangle ) lie in the same plane or not. Show Solution
So, as we noted prior to this example all we need to do is compute the volume of the parallelepiped formed by these three vectors. If the volume is zero they lie in the same plane and if the volume isn’t zero they don’t lie in the same plane.
So, the volume is zero and so they lie in the same plane.
Generic Name:loperamide (loe PER a mide) Brand Name:Diamode, Imodium A-D, Anti-Diarrheal [OTC]
Medically reviewed by Sophia Entringer, PharmD. Last updated on May 1, 2019.
What is Imodium A-D?
Imodium A-D (loperamide) slows the rhythm of digestion so that the small intestines have more time to absorb fluid and nutrients from the foods you eat.
Imodium A-D is used to treat diarrhea.
Imodium A-D is also used to reduce the amount of stool in people who have an ileostomy (re-routing of the bowel through a surgical opening in the stomach).
Warnings
You should not use Imodium A-D if you have ulcerative colitis, bloody or tarry stools, diarrhea with a high fever, or diarrhea caused by antibiotic medication.
Do not take more than the recommended dose of this medicine. Misuse of loperamide can cause serious heart problems that could be fatal.
Serious heart problems may also happen if you take Imodium A-D with other medicines. Ask a doctor or pharmacist about safely using medications together.
Loperamide may impair your thinking or reactions. Be careful if you drive or do anything that requires you to be alert.
Imodium A-D should not be given to a child younger than 2 years old. Loperamide can cause serious breathing problems and heart problems in a child.
Drink plenty of liquids while you are taking this medicine. It may take up to 48 hours of taking Imodium A-D before your symptoms improve. Keep using the medication as directed and tell your doctor if your symptoms do not improve after 10 days of treatment.
Before taking this medicine
You should not use Imodium A-D if you are allergic to loperamide, or if you have:
stomach pain without diarrhea;
diarrhea with a high fever;
ulcerative colitis;
diarrhea that is caused by a bacterial infection; or
stools that are bloody, black, or tarry.
Do not give Imodium A-D to a child younger than 2 years old. Do not give this medicine to an older child or teenager without a doctor's advice.
Ask a doctor or pharmacist if it is safe for you to use this medicine if you have:
a fever;
mucus in your stools;
liver disease; or
a heart rhythm disorder.
Ask your doctor before using Imodium A-D to treat diarrhea caused by Clostridium difficile.
Ask a doctor before using this medicine if you are pregnant or breastfeeding.
You should not breast-feed while you are using loperamide.
How should I take Imodium A-D?
Use Imodium A-D exactly as directed on the label, or as prescribed by your doctor.
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Imodium A-D is safe when used as directed. TAKING TOO MUCH LOPERAMIDE CAN CAUSE SERIOUS HEART PROBLEMS OR DEATH.
Carefully follow all dosing directions on the medicine label. A safe dose of loperamide is different for an adult than for a child. This medicine doses in children are based on the child's age.
Take Imodium A-D with a full glass of water. Diarrhea can cause your body to lose fluids and electrolytes. Drink plenty of liquids to keep from getting dehydrated.
The chewable tablet must be chewed before swallowing. Take the chewable tablet on an empty stomach (1 hour before or 2 hours after a meal)
Shake the oral suspension (liquid) before you measure a dose. Use the dosing syringe provided, or use a medicine dose-measuring device (not a kitchen spoon).
Not all liquid forms of this medicine are the same strengths. Carefully follow all dosing instructions for the medicine you are using.
Store at room temperature away from moisture and heat. Do not allow the liquid medicine to freeze.
Stop taking Imodium A-D and call your doctor if you still have diarrhea after 2 days of treatment, or if you also have stomach bloating.
What happens if I miss a dose?
Since Imodium A-D is used when needed, it does not have a daily dosing schedule. Call your doctor if your symptoms do not improve after using this medicine.
What happens if I overdose?
Seek emergency medical attention or call the Poison Help line at 1-800-222-1222. An overdose of loperamide can be fatal.
Overdose symptoms may include fast or irregular heartbeats, or fainting. A person caring for you should seek emergency medical attention if you pass out and are hard to wake up.
What to avoid
Avoid drinking tonic water. It can interact with loperamide and may cause serious heart problems.
Avoid becoming dehydrated by drinking plenty of fluids. Avoid vigorous exercise or exposure to hot weather if you are dehydrated.
Imodium A-D liquid may contain alcohol. Avoid driving or hazardous activity until you know how this medicine will affect you. Your reactions could be impaired.
Imodium A-D side effects
Get emergency medical help if you have signs of an allergic reaction to Imodium A-D (hives, difficult breathing, swelling in your face or throat) or a severe skin reaction (fever, sore throat, burning in your eyes, skin pain, red or purple skin rash that spreads and causes blistering and peeling).
Stop taking this medicine and call your doctor at once if you have:
This is not a complete list of side effects and others may occur. Call your doctor for medical advice about side effects. You may report side effects to FDA at 1-800-FDA-1088.
Imodium A-D side effects(more detail)
What other drugs will affect Imodium A-D?
Sometimes it is not safe to use certain medications at the same time. Some drugs can affect your blood levels of other drugs you take. Ask a doctor or pharmacist about safely using medications together.
Loperamide can cause a serious heart problem with high doses.
Many drugs can affect loperamide. This includes prescription and over-the-counter medicines, vitamins, and herbal products. Not all possible interactions are listed here. Tell your doctor about all your current medicines and any medicine you start or stop using.
More about Imodium A-D (loperamide)
During Pregnancy or Breastfeeding
Consumer resources
Other brands
Diamode, Imotil, Kao-Paverin
Professional resources
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Related treatment guides
Further information
Remember, keep this and all other medicines out of the reach of children, never share your medicines with others, and use Imodium A-D only for the indication prescribed.
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