What is the Following Product Assume

What is the Following Product Assume.

The geometric definition of the cantankerous product is good for understanding the backdrop of the cross production. However, the geometric definition isn’t so useful for computing the cross product of vectors. For computations, nosotros volition want a formula in terms of the components of vectors. We start by using the geometric definition to compute the cross product of the standard unit of measurement vectors.

Cross product of unit vectors

Let $\vc{i}$, $\vc{j}$, and $\vc{grand}$ exist the standard unit of measurement vectors in $\R^iii$. (Nosotros define the cross production only in three dimensions. Note that we are assuming a right-handed coordinate system.)

Applet: The standard unit vectors in three dimensions

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The standard unit vectors in three dimensions.
The standard unit vectors in three dimensions, $\vc{i}$ (green), $\vc{j}$ (blue), and $\vc{chiliad}$ (ruby) are length i vectors that point parallel to the $x$-axis, $y$-axis, and $z$-axis respectively. Moving them with the mouse doesn’t change the vectors, as they always indicate toward the positive management of their respective centrality.

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The parallelogram spanned past whatsoever two of these standard unit vectors is a unit foursquare, which has area ane. Hence, by the geometric definition, the cross product must be a unit vector. Since the cantankerous production must be perpendicular to the two unit of measurement vectors, it must be equal to the other unit vector or the opposite of that unit of measurement vector. Looking at the above graph, yous can use the right-hand dominion to determine the following results. \begin{marshal*} \vc{i} \times \vc{j} &= \vc{k}\\ \vc{j} \times \vc{thou} &=\vc{i}\\ \vc{thousand} \times \vc{i} &= \vc{j} \end{align*} This piddling cycle diagram tin assistance you remember these results.

Cycle of unit vectors

What about $\vc{i} \times \vc{k}$? Past the right-hand rule, it must exist $-\vc{j}$. By remembering that $\vc{b} \times \vc{a} = – \vc{a} \times \vc{b}$, you tin infer that \begin{align*} \vc{j} \times \vc{i} &= -\vc{k}\\ \vc{m} \times \vc{j} &= -\vc{i}\\ \vc{i} \times \vc{k} &= -\vc{j}. \end{align*}

Finally, the cross production of any vector with itself is the zero vector ($\vc{a} \times \vc{a}=\vc{0}$). In item, the cross product of whatever standard unit vector with itself is the null vector.

General vectors

With the exception of the two special properties mentioned above ($\vc{b} \times \vc{a} = -\vc{a} \times \vc{b}$, and $\vc{a} \times \vc{a} = \vc{0}$), we’ll simply assert that the cross production behaves similar regular multiplication. Information technology obeys the following properties:

  • $(y\vc{a}) \times \vc{b} = y(\vc{a} \times \vc{b}) = \vc{a} \times (y\vc{b})$,
  • $\vc{a} \times (\vc{b}+\vc{c}) = \vc{a} \times \vc{b} + \vc{a} \times \vc{c}$,
  • $(\vc{b}+\vc{c}) \times \vc{a} = \vc{b} \times \vc{a} + \vc{c} \times \vc{a}$,

where $\vc{a}$, $\vc{b}$, and $\vc{c}$ are vectors in $R^3$ and $y$ is a scalar. (These properties mean that the cross product is linear.) Nosotros can use these properties, along with the cantankerous product of the standard unit vectors, to write the formula for the cross product in terms of components.

We write the components of $\vc{a}$ and $\vc{b}$ every bit: \begin{align*} \vc{a} = (a_1,a_2,a_3)= a_1 \vc{i} + a_2 \vc{j} + a_3 \vc{thousand}\\ \vc{b} = (b_1,b_2,b_3)= b_1 \vc{i} + b_2 \vc{j} + b_3 \vc{k} \cease{marshal*}

Outset, we’ll assume that $a_3=b_3=0$. (Then, the manipulations are much easier.) Nosotros summate: \begin{align*} \vc{a} \times \vc{b} &= (a_1 \vc{i} + a_2 \vc{j}) \times (b_1 \vc{i} + b_2 \vc{j})\\ &= a_1b_1 (\vc{i}\times\vc{i}) + a_1b_2(\vc{i} \times \vc{j}) + a_2b_1 (\vc{j} \times \vc{i}) + a_2b_2 (\vc{j} \times \vc{j}) \finish{marshal*} Since we know that $\vc{i} \times \vc{i}= \vc{0}= \vc{j} \times \vc{j}$ and that $\vc{i} \times \vc{j} = \vc{k} = -\vc{j} \times \vc{i}$, this quickly simplifies to \begin{align*} \vc{a} \times \vc{b} &= (a_1b_2-a_2b_1) \vc{k}\\ &= \left| \begin{array}{cc} a_1 & a_2\\ b_1 & b_2 \end{array} \correct| \vc{k}. \end{marshal*} Writing the result as a determinant, equally we did in the last step, is a handy fashion to remember the issue.

The general case where $a_3$ and $b_3$ aren’t zero is a bit more complicated. However, information technology’south just a matter of repeating the same manipulations in a higher place using the cantankerous production of unit vectors and the properties of the cross product.

We beginning with by expanding out the product \begin{align*} \vc{a} \times \vc{b} &= (a_1 \vc{i} + a_2 \vc{j} + a_3\vc{k}) \times (b_1 \vc{i} + b_2 \vc{j} + b_3\vc{m})\\ &= a_1b_1 (\vc{i}\times\vc{i}) + a_1b_2(\vc{i} \times \vc{j}) + a_1b_3(\vc{i} \times \vc{g})\\ &\quad + a_2b_1 (\vc{j} \times \vc{i}) + a_2b_2 (\vc{j} \times \vc{j}) + a_2b_3 (\vc{j} \times \vc{k})\\ &\quad + a_3b_1 (\vc{thou} \times \vc{i}) + a_3b_2 (\vc{k} \times \vc{j}) + a_3b_3 (\vc{grand} \times \vc{one thousand}) \end{align*} and so summate all the cross products of the unit vectors \brainstorm{align*} \vc{a} \times \vc{b} &= a_1b_2 \vc{k} – a_1b_3 \vc{j} – a_2b_1 \vc{thousand} + a_2b_3 \vc{i} + a_3b_1 \vc{j} – a_3b_2 \vc{i}\\ &= (a_2b_3-a_3b_2 )\vc{i} – (a_1b_3-a_3b_1) \vc{j} +(a_1b_2-a_2b_1) \vc{one thousand}. \terminate{marshal*} Using determinants, we can write the outcome as \brainstorm{align*} \vc{a} \times \vc{b} &=\left| \brainstorm{assortment}{cc} a_2 & a_3\\ b_2 & b_3 \cease{array} \right| \vc{i} – \left| \begin{array}{cc} a_1 & a_3\\ b_1 & b_3 \stop{assortment} \right| \vc{j} + \left| \begin{assortment}{cc} a_1 & a_2\\ b_1 & b_2 \stop{assortment} \right| \vc{k}. \end{align*}

Looking at the formula for the $iii \times 3$ determinant, we run into that the formula for a cross product looks a lot like the formula for the $3 \times 3$ determinant. If we allow a matrix to accept the vector $\vc{i}$, $\vc{j}$, and $\vc{thou}$ as entries (OK, maybe this doesn’t make sense, but this is but every bit a tool to remember the cantankerous product), the $3 \times 3$ determinant gives a handy mnemonic to remember the cross production: \begin{marshal*} \vc{a} \times \vc{b} = \left| \begin{array}{ccc} \vc{i} & \vc{j} & \vc{one thousand}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \stop{array} \right|. \end{align*} This is a compact way to remember how to compute the cross product.

What is the Following Product Assume

Source: https://mathinsight.org/cross_product_formula