If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:4:53

Scalar multiplication: magnitude and direction

Video transcript

all right so we're told that vector V is equal to X comma y so they're not giving us the actual numbers for the X component and the y component they're just saying the X component is going to be the variable x and the y component is going to be the variable y and the magnitude of vector V is going to be equal to Phi all right I think I can digest all of that and it says fill in the blanks to complete the following sentences so they're saying vector vector W right over here and they didn't have to actually write the word vector they have an arrow on top of the W so we know it's a vector so vector W is its components are 3 x + 3 y so it's X component is 3 times the X component of vector V and it's Y component is 3 times the Y component of vector V instead of having a Y it has a 3y stead of having an X it has a 3x so W is essentially vector V being multiplied by the scalar 3 so it's magnitude is going to be 3 times the magnitude of vector V so if vector V has a magnitude of 5 vector W is going to have a magnitude of 15 it's going to have 3 times the magnitude all right now they tell us vector Z is equal to so it's X component is negative 2 times the X component of vector V and it's Y component is negative 2 times the Y component of vector V so you could view vector Z as negative 2 times vector V or the scalar negative 2 times the vector V and so you might say okay well that means its magnitude is going to be negative 2 times the magnitude of V but you have to remember magnitude that's always going to be positive you can one way to think when we talk about even absolute value what you really think about the magnitude of something we're talking about how long it is we're not thinking about its direction so we wouldn't think about the the the negative there if we think about its direction this negative is going to flip its direction and we can think about that in a second but if we just think about magnitude we care about the two so it's going to have two times the magnitude of vector V so instead of five it's going to be 5 times 2 which is 10 so now they say match each vector defined to a vector that could be its equivalent so if we think about it the vector with the smallest magnitude is going to be vector V the other two are we're multiplying vector V by scalars with an absolute value of more than 1 so they're going to have larger magnitudes so the smallest magnitude is this one right over here this is the shortest vector so that would be our vector V now vector W would be one that has 3 times the magnitude and it's going in the same direction so this one has 3 times the magnitude it looks like 3 times roughly and it's definitely going in the same direction and then vector Z would have 2 times the magnitude but since we have we're multiplying it times a negative scalar it's going to go in the opposite direction and this looks like it this purple arrow looks like it's about twice the length of this red arrow and it's going in the opposite direction so I feel pretty good about what we have done here let's do one more of these so let's let's do this one here so here we have a vector V again it has a magnitude of 10 and it says fill in the blanks to complete the following sentences vector W now so now vector W it has its it's taking each of the components of vector V and multiplying them by 1/5 so you can view a vector W as being equal to 1/5 the scalar 1/5 times vector V and so the magnitude of W is going to be 1/5 the magnitude of V so 1/5 of 10 is just going to be 2 now vector Z it looks like it's 3/5 times vector V we're multiplying each of the components times 3/5 so that's like multiplying the entire vector by the scalar 3/5 so it's magnitude is going to be 3/5 the magnitude of vector V and so 3/5 of 10 is 6 all right then they say match each vector defined above to a vector that could be its equivalent so here the the vector the largest magnitude is vector V so this would be vector V and the vector with the smallest magnitude is vector W it's one-fifth the magnitude so this has a magnitude should be ten this should be one-fifth of that which is a magnitude of two and it looks like that and then this should be three-fifths of the ten which is a magnitude of six and that looks about right so I feel good with what we have filled out so hopefully that gives you a sense of how to tackle these types of types of example problems