In the above diagram, the position vector of the particle when it is at point P is the vector OP and when it is at point Q, it is OQ. It's essential to first determine the coordinates of a point, before finding the position vector of that point. If we know the position of any point in the xy-plane, then we can use a formula to determine a position vector between those two points.
Solution: If two points are given in the xy-coordinate system, then we can use the following formula to find the position vector PQ:. Where x 1 , y 1 represents the coordinates of point P and x 2 , y 2 represents the point Q coordinates. Thus, by simply putting the values of points P and Q in the above equation, we can find the position vector PQ:.
Thus, the position vector PQ is equivalent to a vector that starts at the origin. This vector is directed to a point 9 units to the right along the x-axis, and 5 units upward along the y-axis.
Solution: If two points are given in the xy-coordinate system, then we can use the following formula to find the position vector QP:. Note that the position vector QP represents a vector directed from point Q towards point P.
It is different from the position vector PQ, which is directed from P to Q. Thus, by simply putting the values of points P and Q in the above equation, we can find the position vector QP:.
Thus, the position vector QP is equivalent to a vector that starts at the origin. This vector is directed to a point of 9 units that is to the left along the x-axis, and 5 units downward along the y-axis.
Whereas, the displacement vector helps us to find the change in the position vector of a given object. A treasure hunter finds one silver coin at a location What are the polar and rectangular coordinates of these findings with respect to the well?
We use Figure to find the x — and y -coordinates of the coins. To specify the location of a point in space, we need three coordinates x , y , z , where coordinates x and y specify locations in a plane, and coordinate z gives a vertical position above or below the plane. Three-dimensional space has three orthogonal directions, so we need not two but three unit vectors to define a three-dimensional coordinate system.
The order in which the axes are labeled, which is the order in which the three unit vectors appear, is important because it defines the orientation of the coordinate system. The order in which these unit vectors appear defines the orientation of the coordinate system. The order shown here defines the right-handed orientation. A vector in three-dimensional space is the vector sum of its three vector components Figure :.
Magnitude A is obtained by generalizing Figure to three dimensions:. This expression for the vector magnitude comes from applying the Pythagorean theorem twice.
Note that when the z -component is zero, the vector lies entirely in the xy -plane and its description is reduced to two dimensions. During a takeoff of IAI Heron Figure , its position with respect to a control tower is m above the ground, m to the east, and m to the north.
One minute later, its position is m above the ground, m to the east, and m to the north. What is the magnitude of its displacement vector? We take the origin of the Cartesian coordinate system as the control tower. Explain why a vector cannot have a component greater than its own magnitude.
If one of the two components of a vector is not zero, can the magnitude of the other vector component of this vector be zero? If two vectors have the same magnitude, do their components have to be the same? Suppose you walk How far are you from your starting point? What is your displacement vector? What is the direction of your displacement? You drive 7. A sledge is being pulled by two horses on a flat terrain. Find the magnitude and direction of the pull.
A trapper walks a 5. Determine the east and north components of her displacement vector. How many more kilometers would she have to walk if she walked along the component displacements? What is her displacement vector? What are its Cartesian coordinates? Determine their Cartesian coordinates and the distance between them in the Cartesian coordinate system. Round the distance to a nearest centimeter. A chameleon is resting quietly on a lanai screen, waiting for an insect to come by.
If its coordinates are 2. Two points in the Cartesian plane are A 2. Find the distance between them and their polar coordinates. A fly enters through an open window and zooms around the room. In a Cartesian coordinate system with three axes along three edges of the room, the fly changes its position from point b 4. What is its magnitude? Privacy Policy. Skip to main content. Search for:. Distinguish between the vector components of a vector and the scalar components of a vector. Explain how the magnitude of a vector is defined in terms of the components of a vector.
Identify the direction angle of a vector in a plane. Points correspond to vectors that start at the origin, but we may need vectors that start at other points. In spirit they are different things. But the usual convention is to think of vector in the plane or in three-dimensional space as starting at the origin. In that case, a vector is identified precisely by its ending point, giving you an identification between points and vectors.
One way to see that they are different things even if identified in many circumstances , is that you can add vectors, while the sum of points makes no sense. Same with the dot and cross products. What exactly is a vector? So it is the same as a point when you consider it as an element of a set.
Now if you want to talk about cross products and magnitudes, then it becomes a question about linguistics. But bare in mind that you will also cause confusion by doing this. And with doing math, we want to communicate clearly and so Maybe it would be better to say this: Is the vector space the same as a set? Yes, a vector space is a set. But it is also more than a set. We can't add elements of a set, but we can add elements of a vector space because with a vector space you get the definition of an addition.
So in this sense, a point and vector are very much different. So how do you so it. Point usually refers to topological structure. In a set with both structures - a vector space with a specified topology - the context of the argument tends to determine which word is used. This is a question which causes a lot of confusion and it's good that you are trying to clear it up as early as possible. It is clearly a question about the geometric meaning of vectors, so IMHO it is not helpful when people start to involve vector spaces in the discussion.
Let me make the assumption that you know what a point is, and that the confusion begins when vectors are introduced. I don't know how to include diagrams in a post so I must ask you to draw your own.
The usual understanding is that a vector is specified by its length and direction, and not by where it is located in the plane. We often use language a bit loosely and refer to a point as a vector. In this case we mean the vector from the origin to the stated point.
As you can see, a vector from the origin and the point it represents are the same numerically , but they are different conceptually and it's worth spending some time trying to get your head around it.
Another example - if you haven't seen this yet I expect you soon will. Vectors and points are two different things and should not be confused. They both share certain similarities, which makes the transformation of one into another very easy, but they are used in different ways as well as describe different mathematical objects. A point is a location in a coordinate system, that is a location defined relatively to an origin.
If you were to move the origin without moving the point, then the coordinates of the point would change. A vector is a more general object. If you were to move the origin, the components of the vector would not change. You can also think of a vector as a transformation. It can be applied anywhere and have the same effect: displacing a point of a certain distance in a precise direction.
Only then are the vector and the point somewhat equivalent. You can understand that even though it is sometimes useful to represent a point as a vector, you should usually not represent a vector as a point. Even though this is math. Hopefully that will help you to tackle the formal mathematical side on your own. I assume that you're thinking of a point as some position or place in space, which is fine to start with.
Now imagine an arrow going from the origin of space to this point. That defines a vector. Every point in space will have a unique such vector, so there's no ambiguity if I said something like "the vector for the point P". But it works the other way, too, in that if I take any arrow that starts at the origin, it is associated with a unique point in space the tip of the arrow.
So there's no ambiguity if I say, "the point for the vector V". Since there's a one-to-one dictionary between vectors and points in space, we can go ahead and just define a point in terms of its vector, which is what I suspect is being done in the reference you're using. That's the intuition, but as you can see from other comments and answers here, I'm leaving out technical detail and I refer you to them for the niceties. An origin-based vector the only kind that has only "direction and magnitude", as you write can be represented by the point at its head.
So there's a natural correspondence between points and vectors in this context, which is easiest to see if you throw in Cartesian coordinates so that both are expressible as a tuple of real numbers. Just because one can represent the other does not mean that they are the same thing, however. You can talk about the distance between points, but sum and dot product are only meaningful for vectors.
If you apply them to a "point", you're really just treating it as standing in for the corresponding vector. Conversely, points pop up in a lot of places where you cannot form a vector space. What distinguishes them are the operations we define on them. Well, there is a huge technical difference. Although, the conceptual difference really depends on how one tries to visualize things, and that for which we need to visualize them. In some subjects, such as calculus, I generally imagine points and vector ad libidum dots and arrows floating around graphs of functions.
This is because there's typically in beginner calculus an implicit convention to use standard unit vector coordinates. In geometry though the heavy technical difference almost calls for a conceptual difference. Additionally, in other geometries the interaction of points and vectors may be different. Imagine a ball. A series of coordinates can be used to describe the points on this ball.
Similarly, at any given point, a series of components can be used to describe directions vectors in the vicinity of a point. Both use a collection of numbers to describe, so it may not be obvious how they are different, but vectors can be added or subtracted, multiplied by scalars, and so on. The vector space structure is essential to the algebra of directions: you can add any two directions to get a third, for instance.
Is it meaningful to add points? You can add coordinates of points, and you may or may not end up with another point.
It's not obvious that two different ways of assigning coordinates to points would allow points to be "added" in a way that gives the same result for both systems--in fact, I do not believe this to be true. So, with all that having been said to describe the difference between points and vectors, why do we sometimes assign vectors to points? Well, sometimes you can do this meaningfully--when space is "flat," so to speak.
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