Cross-product of two vectors also known as "Vector Product" is a way to multiply two vectors to get a new vector. When we find the cross product of two vectors, the result is always a vector that points in a direction perpendicular (or 90 degrees) to both of the original vectors. This only works in 3D space.
In this article, we will understand the meaning of cross product, its definition, the formula of the cross product, the cross product of perpendicular vectors, the cross product of parallel vectors, the right-hand rule cross product, and the properties of the cross product.
Cross Product of Two Vectors
The cross product of two vectors provides a vector that is perpendicular to the plane formed by the original vectors, with a direction determined by the right-hand rule and a magnitude equal to the area of the parallelogram formed by the two vectors.
The cross product, also known as the vector product, is a binary operation that takes two vectors in a three-dimensional Euclidean space and produces another vector. This operation, denoted by the symbol "×," is useful for calculating the area of a parallelogram formed by the two vectors.
\vec A \times \vec B = |A||B|sin\theta
Direction of Resultant Vector of Cross Product
Direction of the resulting vector is determined by the right-hand rule, which means " if you point the fingers of your right hand in the direction of the first vector and curl them towards the second vector, your thumb points in the direction of the cross-product".
This concept is fundamental in various fields, including physics and engineering, where it helps determine perpendicular vectors and areas in three-dimensional space. Cross product of two independent vectors, A and B, denoted as A × B, yields a new vector that is perpendicular to both A and B and is normal to the plane containing these vectors.
Magnitude of Resultant Vector of Cross Product
Magnitude of the cross product is given by the product of the magnitudes of A and B and the sine of the angle θ between them.
\vec A = A\hat i + A\hat j + A\hat k \\
\vec B = B\hat i + B\hat j + B\hat k
Symbolically, if A = |A| and B = |B|, the cross product is expressed as:
\vec{A} \times \vec{B} = |\vec{A}| \cdot |\vec{B}| \cdot \sin \theta ~\hat{n}
Where
|\vec A| = \sqrt{ A_i ^2 + A_j^2 + A_k^2}\\ |\vec B| = \sqrt{ B_i ^2 + B_j^2 + B_k^2}
If two vectors lie in the X-Y plane, their cross-product results in a vector along the Z-axis, perpendicular to the XY plane. The symbol "×" is used between the original vectors, and the resultant vector is represented as (\vec{c}) in the equation (\vec{a} \times \vec{b} = \vec{c}).
Cross-ProductFormula for cross product of two vectors (\vec{A}) and (\vec{B}) in three-dimensional space is:
\vec A = A\hat i + A\hat j + A\hat k \\ \vec B = B\hat i + B\hat j + B\hat k
\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_i & A_j & A_k \\ B_i & B_j & B_k \end{vmatrix}
A × B = A_i B_i(0) + A_i B_j(k) + A_i B_k(-j) + A_j B_i(-k) + A_j B_j(0) + A_j B_k(i) + A_kB_i(j) + A_k B_j(-i) + A_kB_k(0) = (A_jB_k – A_kB_j)i + (A_kB_i – A_iB_k)j + (A_iB_j – A_jB_i)k
This formula involves finding the determinant of a 3x3 matrix, where the first row represents the unit vectors (\hat{i}), (\hat{j}), and (\hat{k}), the second row represents the components of vector (\vec{A}), and the third row represents the components of vector (\vec{B}).
Matrix Representation of Vector Product
Cross product of two vectors can also be represented using a matrix.
Let (\vec{A} = ❬ a1, a2, a3 ❭) and (\vec{B} = ❬ b1, b2, b3 ❭) be two vectors. Then, the cross product (\vec{A} \times \vec{B}) can be represented as:
\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}
Expanding the determinant:
\vec{A} \times \vec{B} = (\hat{i} · (a2 · b3 - a3 · b2) - (\hat{j} · (a1 · b3 - a3 · b1) + (\hat{k} · (a1 · b2 - a2 · b1)
= (\hat{i} · (a2b3 - a3b2) - (\hat{j} · (a1b3 - a3b1) + (\hat{k} · (a1b2 - a2b1)
= ❬ (a2b3 - a3b2), -(a1b3 - a3b1), (a1b2 - a2b1) ❭
Cross Product of Perpendicular Vectors
Cross product of two perpendicular vectors, the resulting vector is perpendicular to both input vectors and its magnitude is equal to the product of the magnitudes of the input vectors.
Mathematically, if the vectors (\vec{a}) and (\vec{b}) are perpendicular, their cross product is:
\vec{a} \times \vec{b} = |\vec{a}|·|\vec{b}|·sin θ·\vec{n}
Where,
- |\vec{a}| and |\vec{b}| are the magnitudes of (\vec{a}) and (\vec{b}) respectively
- (θ) is angle between the two vectors
- (\vec{n}) is a unit vector perpendicular to the plane formed by (\vec{a}) and (\vec{b})
For example, two perpendicular vectors (\vec{a} = ❬ 1, 0, 0 ❭) and (\vec{b} = ❬ 0, 1, 0 ❭).
Their cross product is:
\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{vmatrix}
= (\hat{i} · (0 · 0) - \hat{j} · (1 · 0) + \hat{k} · (1 · 0))
= \hat{i} · 0 - \hat{j} · 0 + \hat{k} · 0
= ❬0, 0, 0❭
The resulting vector (❬ 0, 0, 0 ❭) is a zero vector, indicating that it is perpendicular to both (\vec{a}) and (\vec{b}), as expected when dealing with perpendicular vectors.
Cross Product of Parallel Vectors
Cross product of parallel vectors is always zero. This is because parallel vectors lie on the same line or are collinear, and the angle between them is either 0 or 180 degrees. In both cases, the sine of the angle is zero, resulting in a cross product of zero. Mathematically, if (\vec{A}) and (\vec{B}) are parallel vectors, then:
\vec{A} \times \vec{B} = |\vec{A}| \cdot |\vec{B}| \cdot \sin \theta = 0
where (θ) is the Angle Between Vectors
Therefore, the cross product of parallel vectors is the zero vector. When computing the cross product of parallel vectors, the resulting vector is a zero vector. This implies that its magnitude is zero.
For example, consider two parallel vectors (\vec{a} = ❬ 1, 2, 3 ❭) and (\vec{b} = ❬ 2, 4, 6 ❭). Their cross product is:
\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 2 & 4 & 6 \end{vmatrix} = \langle 0, 0, 0 \rangle
This result shows that the cross product of parallel vectors is a zero vector, indicating that the vectors are parallel and do not create a plane.
Right-Hand Rule for Cross Product
Right-hand rule is a convention used to determine the direction of the resulting vector when performing a cross product.
- Align your right hand's index finger in the direction of the first vector (\vec{A}) and your middle finger in the direction of the second vector (\vec{B}).
- Your thumb will then point in the direction of the resulting vector (\vec{A} \times \vec{B}).
This convention ensures a consistent way to determine the direction of the resulting vector, especially when dealing with physical quantities like torque or magnetic fields.
If vector (\vec{A}) points towards the north and vector (\vec{B}) points towards the east, then the resulting vector (\vec{A} \times \vec{B}) will point upwards according to the right-hand rule.
Triple Cross Product
Triple cross product is a mathematical operation involving three vectors in three-dimensional space. Given three vectors (\vec{A}), (\vec{B}), and (\vec{C}), the triple cross product is denoted as (\vec{A} \times \vec{B}) \times \vec{C}).
To compute the triple cross product, we first calculate the cross product of vectors (\vec{A}) and (\vec{B}), resulting in a new vector. Then, we take the cross product of this new vector and vector (\vec{C}).
Mathematically, the triple cross product can be represented as:
(\vec{A} \times \vec{B}) \times \vec{C} = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{A} \cdot \vec{B})\vec{C}
This operation is used in various mathematical and physical contexts, such as in rotational dynamics, electromagnetism, and fluid mechanics, where it helps determine angular momentum, magnetic fields, and vorticity, respectively.
Properties of Cross Product
The various properties of cross product are listed below:
Orthogonality: The cross product of two non-zero vectors, (\vec{A}) and (\vec{B}), results in a vector (\vec{C}) that is perpendicular to both (\vec{A}) and (\vec{B}).
Magnitude: The magnitude of the cross product, denoted as (|\vec{A} \times \vec{B}|), is given by the product of the magnitudes of (\vec{A}) and (\vec{B}) multiplied by the sine of the angle (θ) between them:
|\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin \theta
Direction: The direction of the cross product is determined by the right-hand rule, ensuring that the resultant vector is orthogonal to the plane formed by (\vec{A}) and (\vec{B}).
Anticommutativity: The cross product is anticommutative, meaning (\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})).
Linearity: The cross product follows the distributive law and is linear, satisfying (\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}).
Zero Cross Product for Parallel Vectors: If (\vec{A}) and (\vec{B}) are parallel (or collinear), then (\vec{A} \times \vec{B} = \vec{0})
Cross product of the unit vectors:
- \vec{i}×\vec{i}~=~\vec{j}×\vec{j}~=~\vec{k}×\vec{k}~=~0
Cross product of the unit vectors:
- \vec{i}×\vec{j}~=~\vec{k}\\\vec{j}×\vec{k}~=~\vec{i}\\\vec{k}×\vec{i}~=~\vec{j}
Cross Product Vs Dot Product
Cross product and dot product are two types of product operation defined on vectors with each resulting in vector and scaler output respectively. Some of the common differences between cross and dot product are:
| Aspect | Cross Product | Dot Product |
|---|
| Definition | Vector operation resulting in a vector perpendicular to both operands. | Scalar operation resulting in a scalar quantity. |
| Result | Vector | Scalar |
| Direction | Perpendicular to the plane containing the two operands. | Depends on the angle between the two vectors. |
| Magnitude | Magnitude is the product of the magnitudes of the vectors and the sine of the angle between them. | Magnitude is the product of the magnitudes of the vectors and the cosine of the angle between them. |
| Commutative Property | No | Yes |
| Distributive Property | Yes | Yes |
| Associative Property | No | No |
| Applications | Calculating torque, area of parallelogram, magnetic field in electromagnetism. | Calculating work, projections, angle between vectors. |
Application of Cross Product of Two Vectors
Application of cross product are as follow:
- Torque calculation in physics and engineering, where it's used to determine rotational force.
- Magnetic field calculation in electromagnetism, where it helps find the direction of magnetic fields around current-carrying wires.
- Angular momentum calculation in rotational motion problems, indicating the rotational motion's intensity.
- Essential in vector algebra for determining perpendicular vectors and calculating areas of parallelograms.
- Crucial in computer graphics for determining surface normals, aiding in realistic lighting effects in 3D rendering.
Results on Cross Product of Two Vectors
- Area of a Parallelogram with adjacent sides \vec{a} and \vec{b} is |\vec{a}×\vec{b}|
- Area of a Triangle with adjacent sides \vec{a} and \vec{b} is 1/2|\vec{a}×\vec{b}|
Related Article on Cross Product of Two Vectors:
Examples on Cross Product of Two Vectors
Example 1: Calculate the cross product of the vectors (\vec{A} = ❬ 2, -1, 3 ❭) and (\vec{B} = ❬ -3, 4, 1 ❭).
Solution:
To calculate cross product of two vectors ( \vec{A}) and ( \vec{B}), denoted as (\vec{A} \times \vec{B}) ,we can use the following formula:
\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}
Given ( \vec{A} = ❬ 2, -1, 3 ❭) and ( \vec{B} = ❬ -3, 4, 1 ❭), we can substitute these values into the determinant:
\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 3 \\ -3 & 4 & 1 \end{vmatrix}
Expanding determinant:
\vec{A} \times \vec{B} = \hat{i} \begin{vmatrix} -1 & 3 \\ 4 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 3 \\ -3 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & -1 \\ -3 & 4 \end{vmatrix}
Now, compute determinants:
\begin{vmatrix} -1 & 3 \\ 4 & 1 \end{vmatrix} = (-1)(1) - (3)(4) = -1 - 12 = -13
\begin{vmatrix} 2 & 3 \\ -3 & 1 \end{vmatrix} = (2)(1) - (3)(-3) = 2 + 9 = 11
\begin{vmatrix} 2 & -1 \\ -3 & 4 \end{vmatrix} = (2)(4) - (-1)(-3) = 8 - 3 = 5
Finally, substitute the determinants into the cross product:
\vec{A} \times \vec{B} = \hat{i}(-13) - \hat{j}(11) + \hat{k}(5)
\vec{A} \times \vec{B} = -13\hat{i} - 11\hat{j} + 5\hat{k}
So, cross product of ( \vec{A}) and ( \vec{B}) is ( \vec{A} \times \vec{B} = ❬ -13, -11, 5 ❭).
Solution:
To determine the area of the parallelogram formed by two vectors ( \vec{A} ) and ( \vec{B} ), we can use the magnitude of their cross product ( |\vec{A} \times \vec{B}| ). The formula for the magnitude of the cross product is:
|\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin(\theta)
Where ( |\vec{A}| ) and ( |\vec{B}| ) are the magnitudes of vectors ( \vec{A}) and ( \vec{B}) respectively, and (θ) is the angle between the two vectors.
Given ( \vec{A} = ❬ 3, 1, -2 ❭) and ( \vec{B} = ❬ 2, -1, 3 ❭), we can compute their magnitudes:
|\vec{A}| = \sqrt{3^2 + 1^2 + (-2)^2} = \sqrt{9 + 1 + 4} = √14
|\vec{B}| = \sqrt{2^2 + (-1)^2 + 3^2} = \sqrt{4 + 1 + 9} = √14
Next, we calculate the cross product \( \vec{A} \times \vec{B} \) and its magnitude:
\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & -2 \\ 2 & -1 & 3 \end{vmatrix}
= \hat{i} \begin{vmatrix} 1 & -2 \\ -1 & 3 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & -2 \\ 2 & 3 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 1 \\ 2 & -1 \end{vmatrix}
= \hat{i}(1 \cdot 3 - (-2) \cdot (-1)) - \hat{j}(3 \cdot 3 - (-2) \cdot 2) + \hat{k}(3 \cdot (-1) - 1 \cdot 2)
= \hat{i}(3 - 2) - \hat{j}(9 + 4) + \hat{k}(-3 - 2)
= \hat{i} + \hat{j} - 5\hat{k}
The magnitude of ( \vec{A} \times \vec{B} ) is:
|\vec{A} \times \vec{B}| = \sqrt{1^2 + 1^2 + (-5)^2} = \sqrt{1 + 1 + 25} = √27
Now, we can calculate the area of the parallelogram:
Area = |\vec{A} \times \vec{B}| = √27
Therefore, the area of the parallelogram formed by the vectors ( \vec{A}) and ( \vec{B} ) is √27 square units.
Example 3: Given that the cross product of two vectors (\vec{A} \times \vec{B}) yields a vector (\vec{C} = ❬ 4, 5, -3 ❭), if the magnitude of (\vec{A}) is (|\vec{A}| = 2) and the magnitude of (\vec{B}) is (|\vec{B}| = 3), find the angle between (\vec{A}) and (\vec{B}).
Solution:
To find the angle between two vectors ( \vec{A} ) and ( \vec{B}) given their magnitudes and the cross product ( \vec{C} = \vec{A} \times \vec{B} ), we can use the formula for the magnitude of the cross product:
|\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin(\theta)
Given that ( |\vec{A}| = 2 ), ( |\vec{B}| = 3 ), and ( \vec{C} = ❬ 4, 5, -3 ❭), we can calculate the magnitude of ( \vec{C}):
|\vec{C}| = \sqrt{4^2 + 5^2 + (-3)^2} = \sqrt{16 + 25 + 9} = √50
Now, we can substitute the given values into the formula for the magnitude of the cross product:
|\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| \cdot \sin(\theta)
√50 = 2 · 3 · sin(θ)
√50 = 6 · sin(θ)
To find (θ), we divide both sides by 6:
\sin(\theta) = \frac{\sqrt{50}}{6}
Now, we can find ( θ ) by taking the inverse sine (arcsine) of both sides:
\theta = \arcsin\left(\frac{\sqrt{50}}{6}\right)
Using a calculator, we find:
\theta \approx \arcsin\left(\frac{\sqrt{50}}{6}\right) \approx 57.19^\circ
Therefore, the angle between vectors ( \vec{A} ) and ( \vec{B} ) is approximately ( 57.19°).
Practice Questions on Cross Product of Two Vectors
Q1: Given two vectors ( \vec{A} = ❬ 1, 2, -3 ❭) and ( \vec{B} = ❬ 2, -1, 4 ❭ ), find ( \vec{A} \times \vec{B} ).
Q2: Find the area of the parallelogram formed by the vectors ( \vec{A} = ❬ 3, -2, 1 ❭) and ( \vec{B} = ❬ 1, 4, -2 ❭).
Q3: If ( \vec{A} \times \vec{B} = ❬ 2, -3, 5 ❭), and ( |\vec{A}| = 4 ) and ( |\vec{B}| = 3 ), find the angle between ( \vec{A} ) and ( \vec{B} ).
Q4: Determine the cross product of the vectors ( \vec{A} = ❬ -2, 5, 1 ❭) and ( \vec{B} = ❬ 3, -1, 2 ❭).
Q5: Given that the cross product of two vectors ( \vec{A} \times \vec{B} ) yields ( \vec{C} = ❬ 4, -3, 2 ❭), and ( |\vec{A}| = 2 ) and ( |\vec{B}| = 3 ), find the magnitude of ( \vec{C} ).
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