David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.
After watching this video, you will be able to explain what scalar multiplication is and complete basic calculations involving the scalar multiplication of vectors. A short quiz will follow.
When Do We Multiply Vectors?
Some numbers in physics have a direction and some don't. You can be cycling down a trail at 3 miles per hour north, but the thermometer you happen to have on your bike (because you love science so much) can't experience a temperature of 32 degrees north. Temperature has no direction. Because of that, we call it a scalar quantity, or a quantity that has magnitude (size), but no direction. The opposite of this is a vector, like velocity, or force, or magnetic field, which is a quantity that has both a magnitude (size) and direction.
There are hundreds of equations in physics, and they contain a mixture of scalars and vectors. Whenever two vectors are multiplied together in one of these equations, like force multiplied by displacement (which is work), or the velocity of a charge multiplied by magnetic field (which is related to the magnetic force), we can multiply them in two different ways: vector multiplication (otherwise known as a cross product) and scalar multiplication (otherwise known as a dot product).
Which one do you do? That depends on the exact situation. Some equations call for a dot product, while others call for a cross product. Generally, if the answer you're looking for is a scalar quantity, it will be a scalar product. But, if the answer you're looking for is a vector quantity, it will be a vector product.
Work is probably the simplest example of a scalar multiplication of vectors. Work is equal to displacement multiplied by force, or in other words, how far an object moves multiplied by the force applied to make it move. Both displacement and force are vectors.
But, if the force was applied at an angle... say, by pushing diagonally down on a broom as it skirts across the floor, we can make the definition of work more specific. We might say that work is equal to the displacement multiplied by the component of the force that acts in the direction of motion. If you're pushing the broom down at an angle, then it's only the part of the force that points along the floor that we're interested in. Whenever that is the case in a physics situation, we're doing scalar multiplication; we're completing a dot product. So in summary, a scalar multiplication is where you multiply one vector by the component of a second vector that acts in the direction of the first vector.
There are two main equations for calculating a dot product. If you have the overall magnitudes and angles of the vector, you use this equation:
Equation for calculating a dot product when you know overall magnitudes and vector angles
So, if you're multiplying vector A by vector B, you take the magnitude of vector A, multiply it by the magnitude of vector B, and multiply that by the cosine of the angle between them. So, this would be like taking your displacement and multiplying it by F cosine theta, the component of the force that acts in the direction of the displacement.
But, what if you're given a quantity in component form? Maybe you don't know the overall magnitude of a vector, but you do know the x and y components of A and the x and y components of B. In that case, you would use the equation below, and it'll work out exactly the same. Just multiply the two x components together and the two y components (and, if you were in 3 dimensions, you would do the same with the two z components), and add them all up.
Equation for calculating a dot product when you do not have the overall magnitude
Aside from work, other examples of scalar products include magnetic potential energy (which is dipole moment multiplied by magnetic field), magnetic flux (magnetic field multiplied by area), and power (force multiplied by velocity).
Okay, now let's go through an example.
Let's say you're dragging your cousin along the street in a wagon. Because of how tall you are, you can't help but pull up at an angle. It would hurt your knees to bend down all the time. If you pull at an angle of 40 degrees to the horizontal with a force of 50 newtons, and the cart moves 8 meters in the positive x-direction, how much work is done in Joules?
Diagram for example 1
Over 79,000 lessons in all major subjects
Get access risk-free for 30 days,
just create an account.
First of all, we write down what we know. The force, F, is 50 newtons, and the displacement is 8 meters. And, as we discussed previously, work is the scalar multiplication of the force vector and the displacement vector. If we take a look at the diagram of the situation (seen above), we'll see that the angle we're given, 40 degrees, also happens to be the angle between these two vectors. So, the angle, theta, is 40 degrees. Then, using our scalar multiplication equation, we just plug the numbers in and solve. 8 multiplied by 50, multiplied by cosine 40, gives us 306 Joules of work. So, that's our answer.
One more example. This time a more abstract one.
If vector A is represented by the equation 3i + 2j, and vector B is represented by the equation 4i + j, what is the scalar product of these vectors?
This time, we have our vectors in component form. For example, vector A is 3 units in the x direction and 2 units in the y direction. So, we can use our other equation for this example, multiplying the x components and the y components and then adding them up. Putting in the specific numbers, we find that the scalar product is equal to 3 multiplied by 4, plus 2 multiplied by 1, which is 14. And so, 14 is our answer.
In physics, there are scalars and vectors. Scalars are just numbers, like the temperature on your bike. But, vectors also have a direction, like your bike's velocity. Whenever two vectors are multiplied together in one of these equations, like force multiplied by displacement (which is work) or velocity of a charge multiplied by magnetic field (which is related to the magnetic force), we can multiply them in two different ways: vector multiplication (otherwise known as a cross product) and scalar multiplication (otherwise known as a dot product). Some equations call for a dot product, while others call for a cross product. With work, your answer is a scalar (it doesn't have a direction), and so this is an example of a scalar product.
A scalar product, in a nutshell, is one vector multiplied by the component of the second vector pointing in the direction of the first vector. So, for the work example, it's displacement multiplied by the component of the force that acts in the direction of the displacement (the direction the object is moving).
The first scalar multiplication equation says to take the magnitude of vector A, multiply it by the magnitude of vector B, and multiply that by the cosine of the angle between them. But, if you're given your two vectors in component form, you'll need the second equation. The component form equation says to multiply the two x components together and the two y components together (and the two z components, if you were in 3 dimensions), and add them all up. And, that's how to calculate a scalar product.
After you've completed this lesson, you'll be able to:
Differentiate between scalars and vectors
Provide two equations to calculate dot products and explain when to use each
Explain what a scalar product is
Work problems using the two equations for dot products
Did you know… We have over 200 college
courses that prepare you to earn
credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the
first two years of college and save thousands off your degree. Anyone can earn
credit-by-exam regardless of age or education level.