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### Course: Precalculus > Unit 6

Lesson 3: Magnitude of vectors# Vector magnitude from components

Sal finds the magnitude of a vector given its components of (5, -3).

## Want to join the conversation?

- So wait, do we always have to assume a vector starts at (0,0) if ve are only given one of its components? (In this case a)(27 votes)
- No. It is convenient to draw vectors starting at the origin, but it is NOT necessary.

5 is just the vector's LENGTH, and -3 is just the vector's HEIGHT. You can draw the vector starting at any point on the graph, but you have to make sure it has a length of 5 and a height of negative 3.

For example: If you drew the vector starting at point (1, 1) then its terminal point would be (6, -2)(40 votes)

- Why does Sal use two bars to indicate magnitude (||a||) instead of one (|a|)? Is there a reason for that or can either way be used interchangeably?(10 votes)
- The convention is to use double bars for vectors and single bars for complex numbers and scalars.(17 votes)

- Can the magnitude of a vector formed by irrational scalars be negative?(3 votes)
- My understanding of MAGNITUDE is that it is the length of the vector and therefore cannot be negative. But its DIRECTION can be positive or negative.(10 votes)

- Are we assuming that the ray a starts from the origin (0, 0)?(2 votes)
- Depends on the context of the question. In our case yes it started at (0,0) but always pay attention to the change in x and y of the vector because vectors can be position anywhere and still have the same magnitude.

Also the vector wouldn't be considered a ray since the vector's length doesn't start at one point then infinitely goes to another.

(I know this question is old this is more for me to get myself to express what I learned)(3 votes)

- Why is the magnitude of a vector denoted as ||x|| rather than |x|?(2 votes)
- To some people using 1 pair of brackets can mean absolute value.(1 vote)

- Why does Sal use the

formula to find the

distance

of the vector?

magnitude(1 vote)- Because the magnitude is the length of the vector. In other words, it's the distance between 2 points.(4 votes)

- isn't the proper way to denote a vector something like: <1,1> and not (1,1)? I've always seen vectors with <> but maybe that's only for unit vectors?(2 votes)
- I've never seen the <x,y> notation however, I have seen the (x,y) row vector or column vector notation (two big brackets with the x on top and y on bottom inside the brackets). The row vector/column vector notation will be used in matrix algebra.(1 vote)

- my teacher always draws the vectors with pointy parentheses. Is there a reason why? They are called angle brackets.(1 vote)
- This is a notational norm. Since you could potentially confuse (x,y) with a coordinate point, using <x,y> simply tells you, "this is a vector" so you just know when you see the brackets.(3 votes)

- how would you find the angle of change of a vector given its graph with rcos(x),rsin(x)?(2 votes)
- You can find the angle of change of a vector by using the inverse sine, inverse cosine and inverse tangent very similary to how you would find the angles in a triagle:

For the inverse sine you would use opposite/hypotenuse which would the y component over the magnitude:`sin^-1((y component)/(magnitude))`

For the inverse cosine you would use adjacent/hypotenuse which would be the x component over the magnitude:`cos^-1((x component)/(magnitude))`

For the inverse tangent you would use the opposite/adjacent which would be the y component over the x component:`tan^-1((y component)/(x component))`

(1 vote)

- Does all vectors have to start at the origin?(2 votes)
- No.And even this problem is just showing its length over both the axes.That is also represented as 5i-3j(1 vote)

## Video transcript

- [Voiceover] Let's do some examples figuring out the magnitude of a vector if we're just given some
information about it. So, one of the simplest cases would be well, if they just told
us the actual components of the vector. So if they said vector a is equal to, let's say five comma negative three, this means that its
x-component is positive five, its y-component is negative three. Well, if we have this,
then the magnitude of a, the magnitude of a is just going to be, and this really just comes
from the distance formula which just comes from
the Pythagorean theorem, the magnitude of a is just going to be the square root of the x-component squared. So let me do that in a different color. So the square root of
the x-component squared, so five squared, plus the y-component squared, so plus negative three squared. And this is going to be equal to the square root of 25, 25 plus nine, plus nine, which is equal to the square root of 34, which is equal to the square root of 34. And if you want to think
about this visually, this is very easy to do just looking at the actual components. But if you want to make sense of this, why this is essentially just
the Pythagorean theorem, we could draw out a quick
coordinate axis right over here. So that's our y-axis. This is our, let's see, I have a y-component of negative three. So let's see. That is our, actually let me
draw it a little bit different. Let me draw it like this. That is our x-axis. And we see its x-component
is positive five, so one, two, three, four, five. That's five there. And its y-component is negative three. So one, two, three. And so this is negative three. And so we can draw this
vector with its initial point. Remember, we can always
shift around a vector as long as we don't change its
magnitude and to direction. We can start it at the origin, and make it go five in the x-direction and negative three in the y-direction, and so its terminal point
will be right over there at the point five comma negative three. And so the vector, the vector, will look like this. And if we want to figure
out the magnitude, that's just the length of this line. And what we can do is just
set up a right triangle where our change, our change in y is this negative three right over here. That is our change in y. And our change in x is this positive five, is that positive five. And so this is a right triangle. Five squared plus, you could just view the absolute value of this side as three, so five squared plus three squared is going to be the hypotenuse squared. Comes straight out of
the Pythagorean theorem.