Vector Cross Product Formula Properties and Solved Examples
In mathematics, there are several important topics that can help students to score good marks in their final exam. One such topic is the cross product of the same vector or the cross product of two parallel vectors. And if you have always wanted to learn more about these topics, then you are in the right place. We will also focus on concepts like right-hand rule cross product and the cross product is zero.
According to experts, cross product can be defined as the binary operation on two vectors in a three-dimensional space. This further results in a sector that is perpendicular to both of the vectors. It should be noted by students that the cross product of two vectors can be calculated by using the right-hand rule.
For students who don’t know what the right-hand rule is, it is defined as nothing but the resultant of any two vectors. These two vectors should be perpendicular to the other two vectors. By using the cross product, one can also find the magnitude of the final resulting vector.
Students should remember that the cross product of two vectors, which is also known as the vector product, A and B is denoted by A × B. also, the resultant vector will be perpendicular to both the vectors named A and B. You can also refer to the image that is attached below to get a good visual representation of this concept.
(Image will be uploaded soon)
When a student is working with vectors, there are certain key points that he or she should remember. We have created a list of those key points and that list is mentioned below.
The cross product of any two vectors will always result in a vector quantity.
In the concept of vector product, the resulting vector will contain a negative sign if the order of the vectors is changed by the student.
The direction of both A and B will always be perpendicular to the plan that contains A and B.
The null vector is the cross product of any two linear vectors.
The Formula of Cross Products
There are several formulas that are related to the chapter on vectors. In this section, we will look at some of the most important vector formulas. Let’s start with the formula of the cross product. If we assume that θ is the angle that exists between any two given vectors, then the formula can be given by:
A × B = AB sin θ
The same formula can also be written as
A × B = ab sin θ n̂
Here, n̂ is the unit vector.
Students should also be familiar with the concept of direction of the cross product. It should be noted that the direction of the cross product of any two non zero parallel vectors, a and b, can be given by using the right-hand thumb rule.
To apply the right-hand rule, simply use your right hand and point your index finger along the vector a. After that, point your middle finger along the vector b. Finally, use the thumb to ascertain the direction of the cross product.
Let’s look at the cross product of two vectors. Students should remember that the cross product of two vectors can be indicated in the following manner:
X × Y = |X| . |Y| sin θ
Now, take any two vectors like × = xi + yj + z and Y = ai + bj + ck
We know that the cross product of these two vectors can be explained in the matrix form, which is also known as the determinant form. This expression is given below.
X × Y = i (yc - zb) - j (xc - za) + k (xb - ya)
After the cross product of two vectors, the next important topic is the triple cross product. As you might have guessed, the product of three vectors is also known as the triple product. It can also be explained as the cross product of a vector with the cross product of any other two vectors.
To arrive at the formula for the triple cross product, we must assume that there are three vectors, which are represented by A, B, and C. These three vectors can be denoted in the following manner.
A × (B × C) = (A . C) B - (A . B) C
(A × B) × C = -C × (A × B) = -(C . B) A + (C . A) B
The last major topic we need to look at is the cross product in spherical coordinates. Students should remember that the resultant vector of the cross product of any two vectors is perpendicular to both vectors. It is also normal to the plane in which the vectors are lying. The same can be represented by using spherical coordinates in a 3-dimensional system or space.
It is also possible to define a vector in a 3-dimensional system or space. This is done as the first radical distance, also known as r. It can also be explained as the distance of a fixed point to the origin, the second point to the polar angle θ, and the third point to the azimuth angle ϕ.
All this information can also be explained in a form of a formula as we know about the transformation from Cartesian to a spherical form. This representation is given below.
X = r sin θ cos ϕ
Y = r sin θ sin ϕ
Z = r cos θ
Fun Facts About Vector Formula Cross Product
Do you know that cross product has several applications in different contexts? For example, the understanding of cross products is used in computational geometry, engineering, and physics. These are very interesting fields and applications that students can pursue further if they are interested in cross products and the field of science in general.
FAQs on Vector Cross Product in Three Dimensional Geometry
1. What is the vector cross product?
The vector cross product of two vectors is a vector that is perpendicular to both original vectors and represents their oriented area. For vectors a and b in three dimensions, the cross product is written as a × b.
- It exists only in 3D space.
- The magnitude equals |a||b| sinθ, where θ is the angle between them.
- The direction is given by the right-hand rule.
2. What is the formula for the cross product of two vectors?
The formula for the cross product of two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is:
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).
- This can also be written using a 3×3 determinant.
- The result is another vector in ℝ³.
- Each component is calculated using pairwise products and subtraction.
3. How do you calculate the cross product step by step?
To calculate the cross product, use the determinant or component formula carefully step by step.
- Let a = (1, 2, 3) and b = (4, 5, 6).
- Compute components:
• First: 2·6 − 3·5 = 12 − 15 = −3
• Second: 3·4 − 1·6 = 12 − 6 = 6
• Third: 1·5 − 2·4 = 5 − 8 = −3 - Result: a × b = (−3, 6, −3).
4. What does the magnitude of the cross product represent?
The magnitude of the cross product equals the area of the parallelogram formed by the two vectors. It is given by |a × b| = |a||b| sinθ.
- If θ = 0° or 180°, then sinθ = 0 and the cross product is zero.
- If θ = 90°, the magnitude is maximized.
- The area of the triangle formed is ½|a × b|.
5. How do you find the direction of the cross product?
The direction of the cross product is determined using the right-hand rule.
- Point your fingers in the direction of vector a.
- Rotate them toward vector b.
- Your thumb points in the direction of a × b.
6. What is the difference between dot product and cross product?
The dot product gives a scalar, while the cross product gives a vector perpendicular to the originals.
- Dot product formula: a · b = |a||b| cosθ.
- Cross product formula: a × b = |a||b| sinθ n̂.
- Dot product measures projection or alignment.
- Cross product measures perpendicular area and direction.
7. When is the cross product equal to zero?
The cross product is zero when the two vectors are parallel or one vector is the zero vector. Since |a × b| = |a||b| sinθ, the result is zero when:
- θ = 0° or 180° (parallel or anti-parallel vectors).
- Either vector has magnitude zero.
8. Can you give an example of a cross product with unit vectors?
Yes, the cross product of standard unit vectors follows simple cyclic rules.
- i × j = k
- j × k = i
- k × i = j
- Reversing order changes sign, e.g., j × i = −k.
9. Is the cross product commutative?
The cross product is not commutative because a × b = −(b × a).
- Swapping the order reverses the direction.
- The magnitude remains the same.
- This property is called anti-commutativity.
10. What are the applications of the cross product?
The cross product is used to find perpendicular vectors, areas, torque, and rotational effects in physics and engineering.
- Area of parallelogram or triangle.
- Finding a normal vector to a plane.
- Calculating torque = r × F in mechanics.
- Determining angular momentum.
