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# Multiplying a vector by a scalar

Watch Sal change the magnitude of a vector by multiplying it by a scalar. Created by Sal Khan.

## Want to join the conversation?

• Is it possible that you can do vectors in 3D?
• If you are asking if a vector can have 3 dimensions, then yes. It can even have more. V = [x, y, z]
• How do you multiply vectors and negevative numbers?
• The same as you do with positive number except you flip the vector so it is going in the opposite direction. You essentially turn the vector 180 degrees.
• how can vectors be used?
• Vectors can be used in almost every branch of physics, and they have an application on almost everything of the real world. For example, vectors help classical mechanics to describe velocity, acceleration, force, or momentum. A car traveling to the east at x speed, a box sliding down a ramp, and a kid running to the farthest corner of a park. Vectors help more physics' branches such as electric and magnetic fields, quantum physics, and general relativity. For example, the force of a magnetic field and its direction, or a particle's position in a vector field.

http://physics.stackexchange.com/questions/14808/uses-of-vectors-in-real-life
• how can we define the matrix of vector which dont starts from the origin?
• I think that you are asking if there is a way to state the starting point for a vector. If that is the case, then I do not believe there is a convention for this. The matrix of the vector only states the motion of the vector from start point to end point. You would have to state the starting point separate from the vector matrix.
• At you said the vector does not have to start at the origin. I was doing practice problems and I had a hard time figuring out what the answer would look like. The answers provided were not at the origin. It is easy to visualize the answer, i think, when the vector is at the origin. Why does it not have to start at the origin? Suggestions?
• Let's say you have to add two vectors and write the result in a matrix form (like Sal writes on the left). What options do you have (assuming you know the parallelogram rule of addition)?

I could think of 3, and only the last one got me to the right answer. Here, have a look: http://i.imgur.com/d5ZbsY0.png

Option 1: They're not tail-to-tail. No idea how the sum vector would look like, right?
Option 2: You found how the sum vector looks like, but you still can't find its coordinates (components) - they have to be measured relative to the origin! (A good question is... why? And the answer is, because people made up vectors and made up rules of adding them)
Option 3: Now we've got it - if tails of both A and B are at (0;0), then we can easily find vector C's components and write it in matrix form. Yay!

Looking back at option 2, you could still, for example, find the length of vector C and a lot of other things. However, as you progress in linear algebra you're going to be a bit disappointed for the lack of graphical representations along the way. Linear algebra can work with N dimensions, not just two - and that's where its beauty lies.
• What's the bracket thing used to indicate vectors? Does it say anything about the direction the vector goes in?
• The 'bracket thing' is a matrix. The top number indicates the horizontal component of the vector and the bottom number the vertical component of the vector. Think of it as the distance in x and the distance in y.

so [2 1 ] (in columns) is 2 in x direction and 1 in the y direction.
• if a vector doesnt start at the origin but is multiplied by a negative scalar, then does the initial point still remain the same or is an inverted mirror vector formed?
• You can start the vector wherever you want. It doesn't matter where. You can keep the initial point or move it somewhere else.
• What are vectors used for? They don't seem very practical to me right now.
• Most of physics is vectors. Example: If you have a car going 60 mph to the north that clearly has both magnitude and direction. Now let's say a 40 mph wind pushes the cart East, after 2 hours how far is the car from its original starting point? You don't have to answer this question, but it'd be very difficult without the idea of vectors.
• How do you know what to multiply by?
• @ Sal multiples [2 1] by 3 because its the number he chose. Just like in any math problem, just multiply by what the problem asks you to.
• what happens, if we change the order of multiplication in vector product ?
• Scalar*Vector=Vector*Scalar,
Vector1 (dot) Vector2 = Vector2 (dot) Vector1,
This means that the commutative property applies to scalar multiplication, and the dot product, however:
Vector1 (cross) Vector2 does not equal Vector2 (cross) Vector1, it actually equals negative Vector2 (cross) Vector1