11.4 Perpendicular Cross Sectionsap Calculus

Section11.4The Cross Product

Volume Cross Sections Calculus
11.4 Perpendicular Cross Sectionsap Calculus Equation
Math Cross Sections
11.4 Perpendicular Cross Sections Ap Calculus Formulas

For this solid, each cross section perpendicular to the x-axis is an equilateral triangle. Find the volume of this solid. Find the volume of this solid. (d) What is the volume of the solid generated by the region R when it is revolved about the line x = –3.
Illustrating the Right Hand Rule of the cross product. Subsection 11.4.2 Applications of the Cross Product. There are a number of ways in which the cross product is useful in mathematics, physics and other areas of science beyond “just” finding a vector perpendicular to two others. We highlight a few here.
From calculus, we know the volume of an irregular solid can be determined by evaluating the following integral: Where A(x) is an equation for the cross-sectional area of the solid at any point x. We know our bounds for the integral are x=1 and x=4, as given in the problem, so now all we need is to find the expression A(x) for the area of our solid.

We introduced the cross product as a way to find a vector orthogonal to two given vectors, but we did not give a proof that the construction given in Definition 11.4.1 satisfies this property. Theorem 11.4.1 asserts this property holds; we leave the verification to Exercise 47.

“Orthogonality” is immensely important. A quick scan of your current environment will undoubtedly reveal numerous surfaces and edges that are perpendicular to each other (including the edges of this page). The dot product provides a quick test for orthogonality: vectors (vec u) and (vec v) are perpendicular if, and only if, (dotp uv=0text{.})

Volume Cross Sections Calculus

Given two non–parallel, nonzero vectors (vec u) and (vec v) in space, it is very useful to find a vector (vec w) that is perpendicular to both (vec u) and (vec vtext{.}) There is an operation, called the cross product, that creates such a vector. This section defines the cross product, then explores its properties and applications.

This definition can be a bit cumbersome to remember. After an example we will give a convenient method for computing the cross product. For now, careful examination of the products and differences given in the definition should reveal a pattern that is not too difficult to remember. (For instance, in the first component only 2 and 3 appear as subscripts; in the second component, only 1 and 3 appear as subscripts. Further study reveals the order in which they appear.)

Let's practice using this definition by computing a cross product.

A convenient method of computing the cross product starts with forming a particular (3times 3) matrix, or rectangular array. The first row comprises the standard unit vectors (vec itext{,}) (vec jtext{,}) and (vec ktext{.}) The second and third rows are the vectors (vec u) and (vec vtext{,}) respectively. Using (vec u) and (vec v) from Example 11.4.2, we begin with:

begin{equation*}begin{matrix} veciamp vecjamp veck 2amp -1amp 43amp 2amp 5end{matrix}end{equation*}

11.4 Perpendicular Cross Sectionsap Calculus Equation

Now repeat the first two columns after the original three:

begin{equation*}begin{matrix} veciamp vecjamp veckamp veciamp vecj 2amp -1amp 4amp 2amp -13amp 2amp 5amp 3amp 2end{matrix}end{equation*}

This gives three full “upper left to lower right” diagonals, and three full “upper right to lower left” diagonals, as shown. Compute the products along each diagonal, then add the products on the right and subtract the products on the left:

begin{equation*}crossp uv = big(-5veci+12vecj+4veck,big) - big(-3veck+8veci+10vecj,big) = -13veci+2vecj+7veck = la -13,2,7ratext{.}end{equation*}

We practice using this method.

Subsection11.4.1Properties of the Cross Product

Math Cross Sections

It is not coincidence that (crossp vu = -(crossp uv)) in the preceding example; one can show using Definition 11.4.1 that this will always be the case. The following theorem states several useful properties of the cross product, each of which can be verified by referring to the definition.

We introduced the cross product as a way to find a vector orthogonal to two given vectors, but we did not give a proof that the construction given in Definition 11.4.1 satisfies this property. Theorem 11.4.4 asserts this property holds; we leave it as a problem in the Exercise section to verify this.

Property 5 from the theorem is also left to the reader to prove in the Exercise section, but it reveals something more interesting than “the cross product of a vector with itself is (vec 0text{.})” Let (vec u) and (vec v) be parallel vectors; that is, let there be a scalar (c) such that (vecv = cvecutext{.}) Consider their cross product:

begin{align*}crossp uv amp = vecu times (cvec u)amp = c(crossp uu) text{ (by Property 3 of knowl{./knowl/thm_cross_prod_prop.html}{text{Theorem 11.4.4}}) }amp = vec 0 text{ (by Property 5 of knowl{./knowl/thm_cross_prod_prop.html}{text{Theorem 11.4.4}}) }text{.}end{align*}

We have just shown that the cross product of parallel vectors is (vec 0text{.}) This hints at something deeper. Theorem 11.3.5 related the angle between two vectors and their dot product; there is a similar relationship relating the cross product of two vectors and the angle between them, given by the following theorem.

Parallel vectors and the cross product.

We could rewrite Definition 11.3.11 and Theorem 11.4.5 to include (vec 0text{,}) then define that (vec u) and (vec v) are parallel if (vec utimesvec v=vec 0text{.}) Since (vec 0cdot vec v =0) and (vec 0times vec v = vec 0text{,}) this would mean that (vec 0) is both parallel and orthogonal to all vectors. Apparent paradoxes such as this are not uncommon in mathematics and can be very useful. (See also the aside in Section 11.2.)

Note that this theorem makes a statement about the magnitude of the cross product. When the angle between (vecu) and (vecv) is 0 or (pi) (i.e., the vectors are parallel), the magnitude of the cross product is 0. The only vector with a magnitude of 0 is (vec 0) (see Property 9 of Theorem 11.2.15), hence the cross product of parallel vectors is (vec 0text{.})

We demonstrate the truth of this theorem in the following example.

Subsection11.4.2Applications of the Cross Product

11.4 Perpendicular Cross Sections Ap Calculus Formulas

There are a number of ways in which the cross product is useful in mathematics, physics and other areas of science beyond “just” finding a vector perpendicular to two others. We highlight a few here.

This application is perhaps more useful in finding the area of a triangle (in short, triangles are used more often than parallelograms). We illustrate this in the following example.

While this application of the Triple Scalar Product is interesting, it is not used all that often: parallelepipeds are not a common shape in physics and engineering. The last application of the cross product is very applicable in engineering.

While the cross product has a variety of applications (as noted in this chapter), its fundamental use is finding a vector perpendicular to two others. Knowing a vector is orthogonal to two others is of incredible importance, as it allows us to find the equations of lines and planes in a variety of contexts. The importance of the cross product, in some sense, relies on the importance of lines and planes, which see widespread use throughout engineering, physics and mathematics. We study lines and planes in the next two sections.

Exercises11.4.3Exercises

In the following exercises, vectors (vec u) and (vec v) are given. Compute (vec utimesvec v) and check this is orthogonal to both (vec u) and (vec vtext{.})

In the following exercises, the magnitudes of vectors (vec u) and (vec v) in (mathbb{R}^3) are given, along with the angle (theta) between them. Use this information to find the magnitude of (vec utimesvec vtext{.})

In the following exercises, find the area of the parallelogram defined by the given vectors.

In the following exercises, find the area of the triangle with the given vertices.

In the following exercises, find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into two triangles.)

In the following exercises, find the volume of the parallelepiped defined by the given vectors.

Best free coding text editor for mac 2017 crack. In the following exercises, find a unit vector orthogonal to both (vec u) and (vec vtext{.})

Answer

Work Step by Step

$({bf u}times{bf v})times{bf w}$ is perpendicular to both $({bf u}times{bf v})$ and ${bf w}$. ${bf u}times{bf v}$ is perpendicular to both ${bf u}$ and ${bf v}$, if neither of them are zero vectors and they are not parallel (which is the degenerate case in which ${bf u}times{bf v}$=${bf 0}$). So $({bf u}times{bf v})times{bf w}$ is parallel to some vector in the plane determined by ${bf u}$ and ${bf v}$. Therefore, $({bf u}times{bf v})times{bf w}$ lies in the plane determined by ${bf u}$ and ${bf v}$. ${bf u}times({bf v}times{bf w})$ is perpendicular to both ${bf u}$ and $({bf v}times{bf w})$. $({bf v}times{bf w})$ is perpendicular to both ${bf v}$ and ${bf w}$, if neither of them are zero vectors and they are not parallel (which is the degenerate case in which ${bf v}times{bf w}$=${bf 0}$). So ${bf u}times({bf v}times{bf w})$ is parallel to some vector in the plane determined by ${bf v}$ and ${bf w}$. Therefore, ${bf u}times({bf v}times{bf w})$ lies in the plane determined by ${bf v}$ and ${bf w}$.