Examples of unit vector in the following topics:
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- In addition to adding vectors, vectors can also be multiplied by constants known as scalars.
- A useful concept in the study of vectors and geometry is the concept of a unit vector.
- A unit vector is a vector with a length or magnitude of one.
- The unit vectors are different for different coordinates.
- The unit vectors in Cartesian coordinates describe a circle known as the "unit circle" which has radius one.
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- Multiplying a vector by a scalar changes the magnitude of the vector but not the direction.
- Similarly if you take the number 3 which is a pure and unit-less scalar and multiply it to a vector, you get a version of the original vector which is 3 times as long.
- Most of the units used in vector quantities are intrinsically scalars multiplied by the vector.
- For example, the unit of meters per second used in velocity, which is a vector, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds.
- In order to make this conversion from magnitudes to velocity, one must multiply the unit vector in a particular direction by these scalars.
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- Figure 3.1 illustrates the basic idea of projecting one vector onto another.
- What we need to do is multiply $\|\mathbf{b}\| \cos \theta$ by a unit vector in the $\mathbf{a}$ direction.
- Obviously a convenient unit vector in the $\mathbf{a}$ direction is $\mathbf{a}/\|\mathbf{a}\|$ , which equals
- So a vector in the $\mathbf{a}$ with length $\|\mathbf{b}\| \cos \theta$ is given by
- Let $\mathbf{a}$ and $\mathbf{b}$ be any two vectors.
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- A Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra.
- Thus the bound vector represented by $(1,0,0)$ is a vector of unit length pointing from the origin along the positive $x$-axis.
- The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.
- For example, the sum of the vectors $(1,2,3)$ and $(−2,0,4)$ is the vector:
- $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are unit vectors along the $x$-, $y$-, and $z$-axes, respectively.
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- A vector function is a function that can behave as a group of individual vectors and can perform differential and integral operations.
- A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.
- The input of a vector-valued function could be a scalar or a vector.
- In terms of the standard unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ of Cartesian 3-space, these specific type of vector-valued functions are given by expressions such as:
- Because vector functions behave like individual vectors, you can manipulate them the same way you can a vector.
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- A vector function covers a set of multidimensional vectors at the intersection of the domains of $f$, $g$, and $h$.
- Also called vector functions, vector valued functions allow you to express the position of a point in multiple dimensions within a single function.
- The input into a vector valued function can be a vector or a scalar.
- In Cartesian form with standard unit vectors (i,j,k), a vector valued function can be represented in either of the following ways:
- This is a three dimensional vector valued function.
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- Consider a vector field $\mathbf{v}$ on $S$; that is, for each $\mathbf{x}$ in $S$, $\mathbf{v}(\mathbf{x})$ is a vector.
- The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
- The flux is defined as the quantity of fluid flowing through $S$ in unit amount of time.
- Based on this reasoning, to find the flux, we need to take the dot product of $\mathbf{v}$ with the unit surface normal to $S$, at each point, which will give us a scalar field, and integrate the obtained field as above.
- where $r$ is the position vector and $\hat{r}$ is a unit vector in radial direction.
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- The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field.
- The curl is a form of differentiation for vector fields.
- If $\hat{\mathbf{n}}$ is any unit vector, the projection of the curl of $\mathbf{F}$ onto $\hat{\mathbf{n}}$ is defined to be the limiting value of a closed line integral in a plane orthogonal to $\hat{\mathbf{n}}$ as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
- Divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point in terms of a signed scalar.
- where $\left|V\right|$ is the volume of $V$, $S(V)$ is the boundary of $V$, and the integral is a surface integral with n being the outward unit normal to that surface.
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- Another way of adding vectors is to add the components.
- For example, a vector with a length of 5 at a 36.9 degree angle to the horizontal axis will have a horizontal component of 4 units and a vertical component of 3 units.
- This new line is the resultant vector.
- The vector in this image has a magnitude of 10.3 units and a direction of 29.1 degrees above the x-axis.
- Vector Addition Lesson 2 of 2: How to Add Vectors by Components
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- Vectors can be used to represent physical quantities.
- Vectors are a combination of magnitude and direction, and are drawn as arrows.
- Because vectors are constructed this way, it is helpful to analyze physical quantities (with both size and direction) as vectors.
- For example, when drawing a vector that represents a magnitude of 100, one may draw a line that is 5 units long at a scale of $\displaystyle \frac{1}{20}$.
- In drawing the vector, the magnitude is only important as a way to compare two vectors of the same units.