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Cross Product of Two Vectors

Cross product of two vectors is the method of multiplication of two vectors. A cross product is denoted by the multiplication sign(x) between two vectors. It is a binary vector operation, defined in a three-dimensional system. The cross product of two vectors is the third vector that is perpendicular to the two original vectors. Its magnitude is given by the area of the parallelogram between them and its direction can be determined by the right-hand thumb rule. The Cross product of two vectors is also known as a vector product as the resultant of the cross product of vectors is a vector quantity. Here we shall learn more about the cross product of two vectors.

1.	Cross Product of Two Vectors
2.	Cross Product Formula
3.	Right-Hand Rule of Cross Product
4.	Cross Product Properties
5.	Triple Cross Product
6.	Cross Product Example
7.	FAQs on Cross Product of Two Vectors

Cross Product of Two Vectors

Cross product is a form of vector multiplication, performed between two vectors of different nature or kinds. A vector has both magnitude and direction. We can multiply two or more vectors by cross product and dot product. 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 cross product of two vectors or the vector product. The resultant vector is perpendicular to the plane containing the two given vectors.

Cross Product Definition

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

cross product of vectors

We can understand this with an example that if we have two vectors lying in the X-Y plane, then their cross product will give a resultant vector in the direction of the Z-axis, which is perpendicular to the XY plane. The × symbol is used between the original vectors. The vector product 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 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 angle between \(\overrightarrow{a}\) and \(\overrightarrow{c}\) is always 90\(^\circ\).i.e., \(\overrightarrow{a}\) and \(\overrightarrow{c}\) are orthogonal vectors.
The angle 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 90°) so that the cross product of the two vectors will be 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 area of the parallelogram that is spanned by the two vectors.

Cross Product Formula

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

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

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 \(\overrightarrow{b}\).

Let us assume that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are two vectors, such that \(\overrightarrow{a}\)= \(a_1\hat i+a_2 \hat j+a_3 \hat k\) and \(\overrightarrow{b}\) = \(b_1 \hat i+b_2 \hat j+b_3 \hat k\) then by using determinants, we could find the cross product and write the result as the cross product formula using matrix notation.

cross product found using determinants

The cross product of two vectors is also represented using the cross product formula as:

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

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

Right-Hand Rule - Cross 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-hand rule. We follow the following procedure to find out the direction of the result of the cross product of two vectors:

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

Check the image given below to understand this better.

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 easily solve all the problems of vector calculations. The properties of the cross product of two vectors are as follows:

The length of the cross product of two vectors \(= \overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta)\).
Anti-commutative property: \(\overrightarrow{a} \times \overrightarrow{b} = - \overrightarrow{b} \times \overrightarrow{a}\)
Distributive property: \(\overrightarrow{a} \times (\overrightarrow{b} + \overrightarrow{c}) = (\overrightarrow{a}\times \overrightarrow{b} )+ (\overrightarrow{a}\times \overrightarrow{c})\)
Cross product of the zero vector: \(\overrightarrow{a}\times \overrightarrow{0} = \overrightarrow{0}\)
Cross 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 cross product of the unit vectors: \(\overrightarrow{i}\times \overrightarrow{i} =\overrightarrow{j}\times \overrightarrow{j} = \overrightarrow{k}\times \overrightarrow{k} = 0\)
\(\overrightarrow{i}\times \overrightarrow{j} = \overrightarrow{k}\\ \overrightarrow{j}\times \overrightarrow{k}= \overrightarrow{i}\\\overrightarrow{k}\times \overrightarrow{i} = \overrightarrow{j}\)
\(\overrightarrow{j}\times \overrightarrow{i} = \overrightarrow{-k}\\ \overrightarrow{k}\times \overrightarrow{j}= \overrightarrow{-i}\\ \overrightarrow{i}\times \overrightarrow{k} = \overrightarrow{-j}\)

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 plane of the given three vectors. If a, b, and c are the vectors, then the vector triple product of these vectors will 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 role 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 opposite ends of the tap. Torque is applied in this case. In vector form, torque is the cross product of the radius vector (from the axis of rotation to the point of application of force) 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 bolt 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 direction 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 cross product of two vectors is given by the right-hand thumb rule 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 zero vector.

☛ Also Check:

Examples of Cross Product of Two Vectors

Example 1: Two vectors have their scalar magnitude as ∣a∣=2√3 and ∣b∣ = 4, while the angle between the two vectors is 60^∘.

Calculate the cross product of two vectors.

Solution:

We know that sin60° = √3/2

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

Answer: The cross product is 12n.
Question 2: Find the cross product of two vectors \(\overrightarrow{a}\) = (3,4,5) and \(\overrightarrow{b}\) = (7,8,9)

Solution:

The cross product is given as,

\(\begin{align}a \times b &=\begin{matrix} \hat i & \hat j & \hat k\\ 3 & 4 & 5\\ 7 & 8 & 9 \end{matrix}\end{align}\)

= [(4×9)−(5×8)] \( \hat {i }\) −[(3×9)−(5×7)]\( \hat {j} \)+[(3×8)−(4×7)] \( \hat {k}\)

= (36−40)\( \hat i\) −(27−35)\( \hat j\) +(24−28) \( \hat k\) = −4\( \hat i\) + 8\( \hat j\) −4\( \hat k\)

Answer: ∴ \(\overrightarrow{a} \times \overrightarrow{b} \) = −4\( \hat i\) + 8\( \hat j\) −4\( \hat k\)
Example 3: If \(\overrightarrow{a}\) = (2, -4, 4) 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+4²) = √36 = 6

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

∣b∣=√(4²+0²+3²) = √25 = 5
As per the cross product formula, we have

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

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

∣c∣=√(−(12)²+10²+16²)

=√(144+100+256)

=√500

=10√5

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

sinθ = 10√5/(5×6)

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 Cross Product of Two Vectors

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FAQs on Cross Product of Two Vectors

What is The Cross Product of Two Vectors?

The cross 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 direction can be determined by the right-hand 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 plane containing the two vectors. The magnitude of the resultant vector is the product of the sin of the angle between 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 two different ways i.e., dot product and cross product. The results in both of these multiplications of vectors are different. Dot product gives a scalar quantity as a result whereas cross product gives vector quantity. The dot product is the scalar product of two vectors and the cross product of two vectors is the vector product of two vectors. The dot product is also known as the scalar product and the cross product is also 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 Find 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 between the two original vectors, sin θ is used because the area of the parallelogram is obtained by the cross product of two vectors.

Is Cross Product of Two Vectors Always Positive?

When the angle between the two original vectors varies between 180° to 360°, 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 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 product 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 any two given vectors by giving the area between those vectors. The cross product formula is given as,\(\overrightarrow{A} × \overrightarrow{B} =|A||B| sin⁡θ\), 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 Cross Product of two Vectors?

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

What Is the Cross Product Formula Using Matrix Notation?

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

How To Use 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 product formula, \((\vec {A × B})=|A||B|\text{Sin⁡}\vec{θ_n}\)

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

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

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

The right-hand thumb rule for the cross-product of two vectors helps to find out the direction of the resultant vector. If we point our right hand in the direction of the first arrow and curl our fingers in the direction of the second, then our thumb will end up pointing in the direction of the cross product of the two vectors. The right-hand thumb rule gives the cross product formula for finding the direction of the resultant vector.

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