If You Take the Cross Product of Two Vectors and Cross It Again What Will Happen

Cantankerous Product of Two Vectors

Cross product of ii vectors is the method of multiplication of 2 vectors. A cross product is denoted by the multiplication sign(x) between two vectors. It is a binary vector functioning, defined in a three-dimensional system. The cross product of ii vectors is the tertiary vector that is perpendicular to the 2 original vectors. Its magnitude is given by the area of the parallelogram between them and its direction tin can be determined by the right-hand thumb dominion. The Cross product of ii vectors is likewise known as a vector product as the resultant of the cross product of vectors is a vector quantity. Hither nosotros shall larn more than nigh the cross product of 2 vectors.

one.	Cantankerous Product of Two Vectors
2.	Cross Production Formula
3.	Right-Hand Rule of Cross Product
4.	Cantankerous Product Properties
5.	Triple Cross Product
vi.	Cantankerous Product Example
seven.	FAQs on Cross Product of Two Vectors

Cross Product of Two Vectors

Cross product is a form of vector multiplication, performed between 2 vectors of different nature or kinds.  A vector has both magnitude and management. We can multiply two or more vectors by cantankerous production and dot production. When two vectors are multiplied with each other and the production of the vectors is likewise a vector quantity, then the resultant vector is called the cross product of ii vectors or the vector product. The resultant vector is perpendicular to the plane containing the 2 given vectors.

Cross Product Definition

If A and B are two independent vectors, then the result of the cantankerous product of these two vectors (Ax B) is perpendicular to both the vectors and normal to the plane that contains both the vectors. Information technology is represented by:
A ten B= |A| |B| sin θ

We can understand this with an example that if we accept two vectors lying in the X-Y plane, then their cross production volition requite a resultant vector in the management of the Z-axis, which is perpendicular to the XY plane. The × symbol is used between the original vectors. The vector production or the cross product of two vectors is shown as:

\(\overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{c}\)

Where

• \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are two vectors.
• \(\overrightarrow{c}\) is the resultant vector.

Cross Product of Ii Vectors Meaning

Use the image shown below and observe the angles between the vectors\(\overrightarrow{a}\) and \(\overrightarrow{c}\) and the angles between the vectors \(\overrightarrow{b}\) and \(\overrightarrow{c}\).

a × b =|a| |b| sin θ.

• The angle between \(\overrightarrow{a}\) and \(\overrightarrow{c}\) is e'er 90\(^\circ\).i.e., \(\overrightarrow{a}\) and \(\overrightarrow{c}\) are orthogonal vectors.
• The bending between \(\overrightarrow{b}\) and \(\overrightarrow{c}\) is always 90\(^\circ\).i.e., \(\overrightarrow{b}\) and \(\overrightarrow{c}\) are orthogonal vectors.
• We can position \(\overrightarrow{a}\) and \(\overrightarrow{b}\) parallel to each other or at an angle of 0°, making the resultant vector a zero vector.
• To get the greatest magnitude, the original vectors must be perpendicular(bending of ninety°) so that the cross product of the two vectors will exist maximum.

Cross Production Formula

Cantankerous production formula betwixt whatever two vectors gives the expanse between those vectors. The cross product formula gives the magnitude of the resultant vector which is the area of the parallelogram that is spanned by the two vectors.

Cross Product Formula

Consider two vectors \(\overrightarrow{a}\)= \(a_1\hat i+a_2 \hat j+a_3 \hat yard\) and \(\overrightarrow{b}\) = \(b_1 \lid i+b_2 \hat j+b_3 \lid grand\). Let θ be the angle formed between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) and \(\hat n\) is the unit vector perpendicular to the aeroplane containing both \(\overrightarrow{a}\) and \(\overrightarrow{b}\). The cross product of the ii vectors is given by the formula:

\(\overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta) \lid n\)

Where

• \(\mid \overrightarrow a \mid\) is the magnitude of the vector a or the length of \(\overrightarrow{a}\),
• \(\mid \overrightarrow b \mid\) is the magnitude of the vector b or the length of \(\overrightarrow{b}\).

Allow united states of america presume that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are 2 vectors, such that \(\overrightarrow{a}\)= \(a_1\chapeau i+a_2 \chapeau j+a_3 \hat k\) and \(\overrightarrow{b}\) = \(b_1 \hat i+b_2 \hat j+b_3 \hat m\) then past using determinants, we could observe the cross product and write the result as the cantankerous product formula using matrix notation.

The cross product of 2 vectors is also represented using the cross product formula equally:

\(\overrightarrow{a} \times \overrightarrow{b} = \chapeau i (a_2b_3-a_3b_2) \\- \hat j (a_1b_3-a_3b_1)\\+ \hat chiliad (a_1b_2-a_2b_1)\)

Note: \( \hat i, \hat j, \text{ and } \hat yard \) are the unit vectors in the direction of 10 axis, y-axis, and z -centrality respectively.

Right-Mitt Rule - Cantankerous Product of Two Vectors

We can find out the direction of the vector which is produced on doing cross product of two vectors by the right-manus dominion. Nosotros follow the post-obit procedure to find out the direction of the result of the cantankerous product of two vectors:

• Align your alphabetize finger towards the direction of the get-go vector(\(\overrightarrow{A}\)).
• Align the center finger in the direction of the 2d vector \(\overrightarrow{B}\).
• Now the thumb points in the direction of the cantankerous product of two vectors.

Cheque the image given below to empathise this improve.

Cross Product of 2 Vectors Properties

The cross-product properties are helpful to understand conspicuously the multiplication of vectors and are useful to hands solve all the problems of vector calculations. The properties of the cantankerous product of two vectors are as follows:

1. The length of the cantankerous product of ii vectors \(= \overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta)\).
2. Anti-commutative property: \(\overrightarrow{a} \times \overrightarrow{b} = - \overrightarrow{b} \times \overrightarrow{a}\)
3. Distributive property: \(\overrightarrow{a} \times (\overrightarrow{b} + \overrightarrow{c}) = (\overrightarrow{a}\times \overrightarrow{b} )+ (\overrightarrow{a}\times \overrightarrow{c})\)
4. Cantankerous product of the zero vector: \(\overrightarrow{a}\times \overrightarrow{0} = \overrightarrow{0}\)
5. Cross product of the vector with itself: \(\overrightarrow{a}\times \overrightarrow{a} = \overrightarrow{0}\)
6. Multiplied past a scalar quantity:\(\overrightarrow{c}(\overrightarrow{a}\times \overrightarrow{b}) = c\overrightarrow{a}\times \overrightarrow{b} = \overrightarrow{a}\times c\overrightarrow{b}\)
7. The cross product of the unit vectors: \(\overrightarrow{i}\times \overrightarrow{i} =\overrightarrow{j}\times \overrightarrow{j} = \overrightarrow{one thousand}\times \overrightarrow{k} = 0\)
   
8. \(\overrightarrow{i}\times \overrightarrow{j} = \overrightarrow{thousand}\\ \overrightarrow{j}\times \overrightarrow{k}= \overrightarrow{i}\\\overrightarrow{1000}\times \overrightarrow{i} = \overrightarrow{j}\)
9. \(\overrightarrow{j}\times \overrightarrow{i} = \overrightarrow{-k}\\ \overrightarrow{k}\times \overrightarrow{j}= \overrightarrow{-i}\\ \overrightarrow{i}\times \overrightarrow{k} = \overrightarrow{-j}\)

Triple Cross Production

The cross product of a vector with the cross production of the other two vectors is the triple cross product of the vectors. The resultant of the triple cross product is a vector. The resultant of the triple cross vector lies in the plane of the given three vectors. If a, b, and c are the vectors, then the vector triple product of these vectors volition exist of the form:

\((\overrightarrow{a}\times \overrightarrow{b}) \times \overrightarrow{c} = (\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b} -(\overrightarrow{b}\cdot \overrightarrow{c}) \overrightarrow{a}\)

Cantankerous Product of 2 Vectors Case

Cantankerous product plays a crucial role in several branches of science and engineering. 2 very bones examples are shown below.

Example ane: Turning on the tap: Nosotros apply equal and opposite forces at the ii diametrically contrary ends of the tap. Torque is applied in this case. In vector course, torque is the cross product of the radius vector (from the axis of rotation to the indicate of awarding of force) and the forcefulness vector.

i.e. \(\overrightarrow{T} = \overrightarrow{r} \times \overrightarrow{F}\)

Example 2: Twisting a commodities with a spanner: The length of the spanner is one vector. Here the direction we apply force on the spanner (to fasten or loosen the bolt) is another vector. The resultant twist management is perpendicular to both vectors.

Important Notes

• The cross product of two vectors results in a vector that is orthogonal to the two given vectors.
• The direction of the cantankerous product of two vectors is given by the right-hand pollex dominion and the magnitude is given by the area of the parallelogram formed by the original two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\).
• The cross-product of two linear vectors or parallel vectors is a goose egg vector.

☛ Also Check:

• Types of vectors
• Vector formulas
• Components of a Vector
• Cross Product Figurer
• Addition of vectors

Examples of Cross Product of 2 Vectors

1. Example 1: Ii vectors accept their scalar magnitude as ∣a∣=2√iii and ∣b∣ = 4, while the angle betwixt the ii vectors is 60^∘.

   Calculate the cross product of two vectors.

   Solution:

   We know that sin60° = √3/two

   The cantankerous product of the two vectors is given past, \(\overrightarrow{a} \times \overrightarrow{b} \) = |a||b|sin(θ)\( \hat n \) = 2√3×4×√iii/2 = 12\( \hat north \)

   Answer: The cross product is 12n.

2. Question 2: Discover the cantankerous product of 2 vectors \(\overrightarrow{a}\) = (3,4,5) and \(\overrightarrow{b}\) = (vii,8,nine)

   Solution:

   The cross product is given as,

   \(\brainstorm{align}a \times b &=\brainstorm{matrix} \hat i & \hat j & \hat k\\ 3 & four & 5\\ 7 & 8 & 9 \stop{matrix}\end{marshal}\)

   = [(4×ix)−(5×8)] \( \hat {i }\) −[(iii×9)−(5×vii)]\( \lid {j} \)+[(3×8)−(4×seven)] \( \hat {k}\)

   = (36−40)\( \hat i\) −(27−35)\( \hat j\) +(24−28) \( \hat k\) = −4\( \hat i\) + eight\( \lid j\) −iv\( \hat k\)

   Reply: ∴ \(\overrightarrow{a} \times \overrightarrow{b} \) = −4\( \chapeau i\) + eight\( \chapeau j\) −4\( \hat g\)

3. Example 3: If \(\overrightarrow{a}\) = (2, -4, four) and \(\overrightarrow{b}\) = (4, 0,3), find the angle between them.

   Solution

   \(\overrightarrow{a}\) = 2i - 4j + 4k

   \(\overrightarrow{b}\) = 4i + 0j +3k

   The magnitude of \(\overrightarrow{a}\) is

   ∣a∣=√(2²+4²+iv²) = √36 = half dozen

   The magnitude of \(\overrightarrow{b}\) is

   ∣b∣=√(iv²+0ⁱⁱ+iii² ) = √25 = 5
   As per the cantankerous product formula, we have

   \(\brainstorm{align}\overrightarrow{a}\times \overrightarrow{b} &= \begin{matrix} \hat i & \hat j & \hat one thousand\\ 2 & -4 & 4\\ four & 0 & 3 \finish{matrix}\\\\&=[(-4\times3) - (four\times0)]\hat i \\ - &[(three\times ii) - (four\times four)] \hat j \\\\ + &[(2\times 0) -(-4\times 4)]\hat k \\\\&= -12\chapeau i + 10 \hat j +16 \hat k \\ \overrightarrow{a}\times \overrightarrow{b} &= (-12, 10, 16) \end{align}\)

   The length of the \(\overrightarrow{c}\) is

   ∣c∣=√(−(12)^two+x^two+16²)

   =√(144+100+256)

   =√500

   =x√5

   \(\brainstorm{marshal}\overrightarrow{a}\times \overrightarrow{b} &= \mid a \mid \mid b \mid \sin\theta\\\sin\theta &= \dfrac{\overrightarrow{a}\times \overrightarrow{b}}{\mid a \mid \mid b \mid}\end{align}\)

   sinθ = ten√five/(5×6)

   sinθ = √5/3

   θ = sin⁻¹(√five/3)

   θ = sin^−one(0.74)

   θ = 48^∘

   Answer: The bending between the vectors is 48^∘.

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Practise Questions on Cantankerous Production of Two Vectors

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FAQs on Cross Production of Two Vectors

What is The Cross Product of Two Vectors?

The cantankerous product of two vectors on multiplication results in the third vector that is perpendicular to the two original vectors. The magnitude of the resultant vector is given by the area of the parallelogram between them and its management can be adamant by the right-mitt thumb dominion. a × b = c, where c is the cross product of the two vectors a and b.

What is The Result of the Vector Cantankerous Product?

When we notice the cross-product of two vectors, we go another vector aligned perpendicular to the aeroplane containing the two vectors. The magnitude of the resultant vector is the product of the sin of the angle betwixt the vectors and the magnitude of the two vectors. a × b =|a| |b| sin θ.

What is Dot Product and Cross Product of Two Vectors?

Vectors can be multiplied in ii dissimilar ways i.e., dot product and cross production. The results in both of these multiplications of vectors are different. Dot production gives a scalar quantity every bit a consequence whereas cross production gives vector quantity. The dot product is the scalar product of 2 vectors and the cantankerous product of two vectors is the vector production of two vectors. The dot product is also known as the scalar production and the cantankerous product is as well known as the vector production. The vector product of ii vectors is given as: \(\overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta) \hat n\), and dot product formula of 2 vectors is given as: \(\overrightarrow{a}. \overrightarrow{b} = |a| |b| \cos(\theta)\)

How to Find Cantankerous Product of Two Vectors?

The cantankerous product of the two vectors is given past the formula: \(\overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta) \hat north\)

Where

• |\(\overrightarrow a\)| is the magnitude or the length of \(\overrightarrow{a}\),
• |\(\overrightarrow b\)| is the magnitude or the length of \(\overrightarrow{b}\)

Why is Cross Product Sine?

Since θ is the angle betwixt the 2 original vectors, sin θ is used because the expanse of the parallelogram is obtained by the cantankerous product of two vectors.

Is Cantankerous Production of Two Vectors E'er Positive?

When the angle betwixt the 2 original vectors varies between 180° to 360°, and then cross product becomes negative. This is because sin θ is negative for 180°< θ <360°.

What is the Difference Between Dot Product and Cross Production of Ii Vectors?

While multiplying vectors, the dot product of the original vectors gives a scalar quantity, whereas the cross product of two vectors gives a vector quantity. A dot product is the production of the magnitude of the vectors and the cos of the angle between them. a . b = |a| |b| cosθ. A vector product is the product of the magnitude of the vectors and the sine of the angle between them. a × b =|a| |b| sin θ.

What Is the Cross Product Formula for Two Vectors?

Cross product formula determines the cross product for whatever two given vectors by giving the expanse between those vectors. The cross product formula is given equally,\(\overrightarrow{A} × \overrightarrow{B} =|A||B| sin⁡θ\), where |A| = magnitude of vector A, |B| = magnitude of vector B and θ = angle between vectors A and B.

How Do You Find The Magnitude of the Cantankerous Product of two Vectors?

The cross product of two vectors is another vector whose magnitude is given by \(\overrightarrow{a} \times \overrightarrow{b} = \hat i (a_2b_3-a_3b_2) \\- \hat j (a_1b_3-a_3b_1)\\+ \hat thou (a_1b_2-a_2b_1)\)

What Is the Cross Production Formula Using Matrix Annotation?

For the two given vectors, \(\overrightarrow{a}\) and \(\overrightarrow{b}\) nosotros can find the cross product by using determinants. For case, \(\overrightarrow{a}\)= \(a_1\hat i+a_2 \lid j+a_3 \chapeau k\) and \(\overrightarrow{b}\) = \(b_1 \chapeau i+b_2 \hat j+b_3 \lid one thousand\) then we can write the result as, \(\overrightarrow{a} \times \overrightarrow{b} = \hat i (a_2b_3-a_3b_2) \\- \hat j (a_1b_3-a_3b_1)\\+ \hat chiliad (a_1b_2-a_2b_1)\)

How To Use Cantankerous Product Formula?

Consider the given vectors.

• Step 1: Check for the components of the vectors, |A| = magnitude of vector A, |B| = magnitude of vector B and θ = angle between vectors A and B.
• Step 2: Put the values in the cross product formula, \((\vec {A × B})=|A||B|\text{Sin⁡}\vec{θ_n}\)

For example, if \(\vec {A}=a\hat{i} + b\hat{j}+c\hat{thousand}\) and \( \vec{B}=d\hat{i} + e\chapeau{j}+f\hat{k}\) so \({\vec{A × B}} = \begin{matrix} \lid{i} & \hat{j} & \chapeau{thou} \\ a & b & c \\ d & e & f \end{matrix}\)

\({\vec{A × B}} = \hat{i}(bf-ce) - \hat{j}(af-cd) + \lid{g}(ae-bd)\)

What Is the Right Hand Thumb Rule for Cross Product of 2 Vectors?

The right-hand thumb rule for the cross-product of two vectors helps to find out the direction of the resultant vector. If nosotros bespeak our right hand in the direction of the offset arrow and gyre our fingers in the direction of the second, then our thumb will finish upwards pointing in the management of the cantankerous product of the ii vectors. The correct-hand thumb dominion gives the cantankerous product formula for finding the management of the resultant vector.

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Source: https://www.cuemath.com/geometry/cross-product/#:~:text=The%20cross%20product%20of%20two%20vectors%20on%20multiplication%20results%20in,the%20right%2Dhand%20thumb%20rule.