Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.
From airplanes to walking in a straight line, vectors are everywhere. In this lesson, we're going to learn how to add, subtract, multiply, and divide vectors helping us make more sense of them.
Quick Review of Vectors
Have you ever been on a plane and wondered what the pilot meant when she mentioned a headwind or a tailwind? Why in the world would such things matter to your arrival time? Or have you ever looked up on a windy day and swore that the birds were actually flying sideways? No, I'm afraid you didn't bump your head. Instead, what you are seeing is the world of vectors at work.
Remember that a vector is a magnitude combined with a direction. Saying we went north does not qualify as a vector. However, saying we walked two miles per hour north does. When graphed, vectors show their magnitude, or overall strength of force, by how long they are. This could often just be the number of miles traveled within a given time frame, often an hour. The direction of a vector is indicated by the direction of the line. In this lesson, we're going to learn how to add, subtract, and multiply vectors.
First things first, relax! Yes, you're going to have vectors going in odd angles from each other, but that doesn't mean that you'll have to learn any advanced trigonometry. In fact, you won't need any trigonometry at all to add vectors. Simply start at the tail of one of the vectors and draw a line to the tip of the other. Since vectors show their magnitude over a given time, this new line represents how the object in question will actually move over the course of that given time.
Say you had a vector with a magnitude of ten going northeast and a vector with a magnitude of four going north. First things first, draw the first vector. It doesn't matter which you choose. Now, at the tip of that vector, draw the tail of the second one. Draw out the vector like you would anywhere else. Now, simply draw a line from the tail of the first vector to the tip of the second vector. That is the result of adding these two vectors! Oddly enough, the resulting vector is called the resultant.
In order to subtract vectors, you do basically the same thing but with one minor adjustment. Instead of drawing the second vector as is, flip it 180 degrees so that it has completely changed direction. In effect, you are adding a negative quantity to the first vector, so it would go in the opposite direction than what is listed. From there, simply carry out the addition of vectors like before.
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Adding and subtracting vectors is pretty straightforward, but neither is as easy as multiplying a vector by a scalar. Later in math you'll learn how to multiply two vectors together, but for right now we just have to know how to use a scalar. Speaking of which, what is a scalar anyway? A scalar is simply a number that we multiply a vector by. For example, if we had a vector that showed us going northwest at 30 miles per hour, a scalar could be used to change the magnitude of that vector. If we wanted to double our speed, we'd use a scalar of two. Simply multiply the scalar by the magnitude to receive the new quantity of the magnitude.
But wait, if you can add and subtract vectors, then shouldn't you be able to divide them if you can multiply them? As a matter of fact, you can. However, just like subtracting the vector involved switching the direction so that it was more like adding a negative number, dividing by a vector really just means multiplying by a scalar that is the inverse of the number to be divided by. For example, if you had a vector of a ship in motion and wanted to slow its speed by half, you would divide by a scalar of two, which is the same as multiplying by a scalar of one half.
In this lesson, we learned how to add, subtract, multiply, and divide vectors. Remember a vector must have a magnitude and a direction. To add or subtract vectors, simply draw the vectors together, making sure to flip the second vector if it is a subtraction problem. Then draw a line from the tail of the first vector to the tip of the second one. For multiplication, we learned to multiply magnitude by scalars. Additionally, for division, we learned to multiply by the inverse of the scalar to get the same effect.
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