Geometric Definition Of Cross Product

Cross Product of Two Vectors

Cross product of two vectors is the method of multiplication of two vectors. A cross product is denoted past the multiplication sign(x) betwixt ii vectors. It is a binary vector operation, defined in a 3-dimensional organization. The cross production of 2 vectors is the tertiary vector that is perpendicular to the ii original vectors. Its magnitude is given by the area of the parallelogram between them and its direction tin be determined by the right-hand thumb dominion. The Cross product of two vectors is also known as a vector product every bit the resultant of the cantankerous product of vectors is a vector quantity. Here we shall learn more than near the cross product of two vectors.

1.	Cross Product of Two Vectors
2.	Cross Product Formula
three.	Right-Paw Rule of Cross Product
4.	Cross Production Properties
5.	Triple Cross Production
6.	Cross Product Instance
7.	FAQs on Cross Production of 2 Vectors

Cantankerous Product of Two Vectors

Cantankerous product is a course of vector multiplication, performed between two vectors of different nature or kinds. A vector has both magnitude and management. We tin multiply two or more vectors by cantankerous product and dot production. When two vectors are multiplied with each other and the product of the vectors is also a vector quantity, then the resultant vector is called the cantankerous product of two vectors or the vector production. The resultant vector is perpendicular to the aeroplane containing the two given vectors.

Cross Production Definition

If A and B are two independent vectors, and then the result of the cross product of these 2 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 x B= |A| |B| sin θ

cross product of vectors

Nosotros can understand this with an example that if we have two vectors lying in the 10-Y plane, then their cross product will give a resultant vector in the direction of the Z-centrality, which is perpendicular to the XY plane. The × symbol is used between the original vectors. The vector product or the cross product of ii vectors is shown equally:

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

Where

\(\overrightarrow{a}\) and \(\overrightarrow{b}\) are 2 vectors.
\(\overrightarrow{c}\) is the resultant vector.

Cantankerous Product of Two 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.

cross product of a and b

The bending betwixt \(\overrightarrow{a}\) and \(\overrightarrow{c}\) is always 90\(^\circ\).i.eastward., \(\overrightarrow{a}\) 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(angle of ninety°) so that the cantankerous product of the two vectors volition exist maximum.

Cross Product Formula

Cross product formula between any two vectors gives the area between those vectors. The cross product formula gives the magnitude of the resultant vector which is the expanse of the parallelogram that is spanned by the two vectors.

Cross Product Formula

Consider ii vectors \(\overrightarrow{a}\)= \(a_1\chapeau i+a_2 \hat j+a_3 \lid k\) and \(\overrightarrow{b}\) = \(b_1 \chapeau i+b_2 \hat j+b_3 \hat k\). Allow θ be the bending formed between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) and \(\chapeau northward\) is the unit of measurement vector perpendicular to the plane containing both \(\overrightarrow{a}\) and \(\overrightarrow{b}\). The cross product of the ii vectors is given by the formula:

\(\overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta) \chapeau due north\)

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 \(\overrightarrow{b}\).

Permit united states assume that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are two vectors, such that \(\overrightarrow{a}\)= \(a_1\lid i+a_2 \hat j+a_3 \chapeau k\) and \(\overrightarrow{b}\) = \(b_1 \lid i+b_2 \hat j+b_3 \hat chiliad\) and then by using determinants, we could detect the cross product and write the result as the cross product formula using matrix note.

cross product found using determinants

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

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

Note: \( \chapeau i, \hat j, \text{ and } \lid k \) are the unit vectors in the direction of x axis, y-axis, and z -axis respectively.

Right-Manus Rule - Cross Product of Two Vectors

We tin can detect out the direction of the vector which is produced on doing cross product of two vectors by the right-manus rule. We follow the following procedure to discover out the direction of the result of the cantankerous production of 2 vectors:

Marshal your index finger towards the direction of the offset vector(\(\overrightarrow{A}\)).
Align the middle finger in the direction of the second vector \(\overrightarrow{B}\).
Now the pollex points in the direction of the cross product of two vectors.

Check the image given beneath to empathize this ameliorate.

Right thumb rule of cross product

Cross Product of Two Vectors Properties

The cross-product properties are helpful to understand clearly the multiplication of vectors and are useful to hands solve all the problems of vector calculations. The properties of the cross product of 2 vectors are equally follows:

The length of the cantankerous product of 2 vectors \(= \overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta)\).
Anti-commutative property: \(\overrightarrow{a} \times \overrightarrow{b} = - \overrightarrow{b} \times \overrightarrow{a}\)
Distributive holding: \(\overrightarrow{a} \times (\overrightarrow{b} + \overrightarrow{c}) = (\overrightarrow{a}\times \overrightarrow{b} )+ (\overrightarrow{a}\times \overrightarrow{c})\)
Cross product of the zero vector: \(\overrightarrow{a}\times \overrightarrow{0} = \overrightarrow{0}\)
Cantankerous product of the vector with itself: \(\overrightarrow{a}\times \overrightarrow{a} = \overrightarrow{0}\)
Multiplied by a scalar quantity:\(\overrightarrow{c}(\overrightarrow{a}\times \overrightarrow{b}) = c\overrightarrow{a}\times \overrightarrow{b} = \overrightarrow{a}\times c\overrightarrow{b}\)
The cantankerous production of the unit vectors: \(\overrightarrow{i}\times \overrightarrow{i} =\overrightarrow{j}\times \overrightarrow{j} = \overrightarrow{grand}\times \overrightarrow{one thousand} = 0\)
\(\overrightarrow{i}\times \overrightarrow{j} = \overrightarrow{m}\\ \overrightarrow{j}\times \overrightarrow{k}= \overrightarrow{i}\\\overrightarrow{chiliad}\times \overrightarrow{i} = \overrightarrow{j}\)
\(\overrightarrow{j}\times \overrightarrow{i} = \overrightarrow{-g}\\ \overrightarrow{k}\times \overrightarrow{j}= \overrightarrow{-i}\\ \overrightarrow{i}\times \overrightarrow{chiliad} = \overrightarrow{-j}\)

Triple Cross Product

The cross product of a vector with the cross product 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 aeroplane of the given three vectors. If a, b, and c are the vectors, and so the vector triple production of these vectors volition be of the form:

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

Cross Product of Two Vectors Example

Cross product plays a crucial function in several branches of science and engineering. Two very basic examples are shown below.

Example 1: Turning on the tap: We apply equal and opposite forces at the two diametrically contrary ends of the tap. Torque is practical in this case. In vector course, torque is the cross product of the radius vector (from the axis of rotation to the point of application of strength) and the force vector.

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

Turning on the tap and the fastening of the screw using the spanner-Torque application

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

Important Notes

The cross product of 2 vectors results in a vector that is orthogonal to the two given vectors.
The direction of the cross product of two vectors is given by the right-hand thumb rule and the magnitude is given by the surface area of the parallelogram formed past the original two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\).
The cross-product of ii linear vectors or parallel vectors is a zero vector.

☛ Also Bank check:

Types of vectors
Vector formulas
Components of a Vector
Cross Product Calculator
Add-on of vectors

Examples of Cross Product of 2 Vectors

Example 1: Two vectors accept their scalar magnitude as ∣a∣=2√3 and ∣b∣ = 4, while the angle betwixt the two vectors is threescore^∘.

Calculate the cross product of 2 vectors.

Solution:

Nosotros know that sin60° = √3/ii

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

Reply: The cross product is 12n.
Question two: Find the cross production of two vectors \(\overrightarrow{a}\) = (iii,4,v) and \(\overrightarrow{b}\) = (7,viii,9)

Solution:

The cross product is given equally,

\(\begin{align}a \times b &=\brainstorm{matrix} \hat i & \hat j & \hat k\\ 3 & four & 5\\ 7 & eight & nine \end{matrix}\end{marshal}\)

= [(four×9)−(5×8)] \( \hat {i }\) −[(3×9)−(5×7)]\( \lid {j} \)+[(three×8)−(four×vii)] \( \lid {chiliad}\)

= (36−xl)\( \chapeau i\) −(27−35)\( \hat j\) +(24−28) \( \hat yard\) = −iv\( \chapeau i\) + eight\( \hat j\) −4\( \hat one thousand\)

Answer: ∴ \(\overrightarrow{a} \times \overrightarrow{b} \) = −four\( \chapeau i\) + 8\( \hat j\) −4\( \hat k\)
Instance 3: If \(\overrightarrow{a}\) = (2, -4, 4) and \(\overrightarrow{b}\) = (4, 0,3), find the angle betwixt them.

Solution

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

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

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

∣a∣=√(2²+4ⁱⁱ+four²) = √36 = 6

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

∣b∣=√(4²+0^two+3ⁱⁱ) = √25 = 5
Every bit per the cantankerous product formula, we accept

\(\begin{align}\overrightarrow{a}\times \overrightarrow{b} &= \begin{matrix} \hat i & \lid j & \lid k\\ 2 & -iv & 4\\ 4 & 0 & 3 \terminate{matrix}\\\\&=[(-four\times3) - (iv\times0)]\lid i \\ - &[(3\times 2) - (4\times iv)] \hat j \\\\ + &[(ii\times 0) -(-4\times iv)]\hat 1000 \\\\&= -12\hat i + 10 \hat j +sixteen \hat m \\ \overrightarrow{a}\times \overrightarrow{b} &= (-12, ten, 16) \cease{marshal}\)

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

∣c∣=√(−(12)²+10²+16ⁱⁱ)

=√(144+100+256)

=√500

=10√5

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

sinθ = 10√five/(five×half dozen)

sinθ = √5/3

θ = sin⁻¹(√5/3)

θ = sin⁻¹(0.74)

θ = 48^∘

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

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Practice Questions on Cantankerous Production of Two Vectors

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FAQs on Cantankerous 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 expanse of the parallelogram between them and its direction tin can exist determined past the right-manus thumb rule. a × b = c, where c is the cross product of the two vectors a and b.

What is The Result of the Vector Cross Product?

When we find the cross-product of two vectors, we get another vector aligned perpendicular to the aeroplane containing the ii 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 Cantankerous Product of Two Vectors?

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

How to Detect Cross Product of Two Vectors?

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

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 two original vectors, sin θ is used considering the area of the parallelogram is obtained by the cantankerous production of two vectors.

Is Cross Product of Ii Vectors E'er Positive?

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

What is the Deviation Between Dot Product and Cross Product of Two 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 production 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 any two given vectors past giving the area between those vectors. The cross production 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 = bending betwixt vectors A and B.

How Exercise Y'all Find The Magnitude of the Cross Product of two Vectors?

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

What Is the Cross Production Formula Using Matrix Note?

For the two given vectors, \(\overrightarrow{a}\) and \(\overrightarrow{b}\) we tin find the cross product past using determinants. For example, \(\overrightarrow{a}\)= \(a_1\chapeau i+a_2 \hat j+a_3 \chapeau thousand\) and \(\overrightarrow{b}\) = \(b_1 \lid i+b_2 \hat j+b_3 \hat 1000\) then nosotros can write the effect as, \(\overrightarrow{a} \times \overrightarrow{b} = \hat i (a_2b_3-a_3b_2) \\- \hat j (a_1b_3-a_3b_1)\\+ \lid k (a_1b_2-a_2b_1)\)

How To Employ Cross 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 production formula, \((\vec {A × B})=|A||B|\text{Sin⁡}\vec{θ_n}\)

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

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

What Is the Right Hand Pollex Rule for Cross Product of Two Vectors?

The right-mitt thumb rule for the cross-product of 2 vectors helps to observe out the direction of the resultant vector. If we bespeak our right hand in the direction of the first arrow and curl our fingers in the direction of the second, then our thumb will finish upward pointing in the direction of the cantankerous product of the two vectors. The right-hand thumb rule gives the cross product formula for finding the direction of the resultant vector.