Examples of cross product in the following topics:
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- The cross product of two vectors is a vector which is perpendicular to both of the original vectors.
- The cross product is a binary operation of two three-dimensional vectors.
- If the two original vectors are parallel to each other, the cross product will be zero.
- The cross product is denoted as $a \times b = c$.
- The cross product is different from the dot product because the answer is in vector form in the same number of dimensions as the original two vectors, where the dot product is given in the form of a single quantity in one dimension.
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- The dot product takes two vectors and returns a single value.
- The dot product can only be taken from two vectors of the same dimension.
- The dot product is the sum of the product of the corresponding parameters.
- Geometrically, the dot product is the product of the magnitudes of two vectors and the cosine of the angle between them.
- This is different from the cross product, which gives an answer in vector form.
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- If you were to take a cross section, with the cut perpendicular to any of the three axes, you would see the graph of that function.
- The vector valued functions can be manipulated in the same way as a vector; they can be added, subtracted, and the dot product and the cross product can be found.
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- 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.
- The cross product on the right-hand side of this expression is a surface normal determined by the parametrization.
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- A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
- In this particular equation, the constant $m$ determines the slope or gradient of that line, and the constant term $b$ determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.
- Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than $1$) or other function of a variable, equations involving terms such as $xy$, $x^2$, $y^{\frac{1}{3}}$, and $\sin x$ are nonlinear.
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- For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2: $2^{3} = 2 \times 2\times 2 = 8$.
- The graph of the logarithm to base 2 crosses the x-axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3).
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- In general, the contour lines of $f$ and $g$ may be distinct, so following the contour line for $g = c$, one could intersect with or cross the contour lines of $f$.
- When the contour line for $g = c$ meets contour lines of $f$ tangentially we neither increase nor decrease the value of $f$—that is, when the contour lines touch but do not cross.
- Where the Lagrange multiplier $\lambda=0$ we can have a local extremum and the two contours cross instead of meeting tangentially.
- Therefore where the constraint $g=c$ crosses the contour line $f=-1$, is a local minimum of $f$ on the constraint.
- The trace and the contour $f=-1$ cross at the minimum as we can see in the figure.
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- If we wanted, we could obtain a general expression for the volume of blood across a cross section per unit time (a quantity called flux).
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- A line is essentially a representation of a cross section of a plane, or a two dimensional version of a plane which is a three dimensional object.
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- Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with a round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, and so on.