Vector Product Formula Steps Properties and Solved Examples
Vector Product
A vector has both the direction which is indicated by an arrow as well as the magnitude which is indicated by the length. A vector product is a combination of two vectors i.e., scalar and vector. Therefore, we have two ways in which we can multiply the vectors. First is the dot product of vectors which is also known as the Scalar product. Another is the cross product of vectors which is also known as the vector product. By the end of this, we will surely be able to define and calculate the vector product when the two vectors are provided in a cartesian form and learn the geographical applications of it.
Define Vector Product
Now, can you define a vector product? You can start with an example. Here are two vectors (a and b) and the angle between them is represented as .
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We, by now, know that when two vectors are multiplied, the result is always a vector. So, to obtain a vector, we will first have to specify the direction. And by the definition, the direction of the vector product is at the right angles to both a and b. This also means that they are at right angles even to the plane in which a and b lies.
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Therefore, we have two choices. To make this choice we can draw help from the right-hand screw rule. According to this rule, the direction of the vector product would be in the same direction as the direction in which the screwdriver would turn, i.e., from a to b.
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The vector product of a and b is to be defined as: a x b = |a||b| sin \[\theta\] \[\widehat{n}\]
Where, |a| is the modulus or the magnitude of a,
|b| is the modulus of b
\[\theta\] is the angle between a and b
\[\widehat{n}\] is the unit of vectors which is perpendicular to both a and b.
Note: Vector product is also called cross vector product as the symbol of vector product is x
Properties Of Vector Product
Before proceeding forward, there are few properties of vector products that we must know. These are:
The order in which we perform the calculation matters, as a x b and b x a, are opposite to each other. Therefore, the vector product is not commutative.
The vector product is always distributive over addition, for example:
a x (b + c) = a x b + a x c
These are the basic vector product properties that will be helpful for you.
Cross Vector Product Of Two Parallel Vectors
Consider the two vectors (a and b) parallel but the definition of vector does not apply to parallel lines as two parallel vectors do not define a plane. Therefore, the vector product of the two parallel vectors will be zero.
Cross Vector Product Of Two Parallel Vectors In Cartesian Form
We can find the vector product of two vectors in a Cartesian form such as a = 3i - 2j + 7k and b = -5i +4j - 3k, where i, j, and k are the unit vectors in the directions of the x, y and z axes respectively. We can use a formula which we will develop in the end. So, first, let us start with a few cross-product examples:
Example 1) consider that we want to find i x j. Now since they lie along the x and y axes, we can say that these vectors are perpendicular.
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Here, we can see that k is the unit vector perpendicular to i and j and the angle between i and j is 90 degree and sin 90 degree is 1. With the help of hand screw rule, we can find i x j. Therefore, i x j = |i||j| sin 900 k
= (1) (1) (1) k
= k
Example 2) Now if we find j x i using the hand screw rule, the vector perpendicular to j and i is equal to -k. Therefore, j x i = -k
Example 3) Finding i x i will result in zero as they are perpendicular and the angle between them is 00. therefore, i x i = 0
Based on these cross product example, we can summarize the following as:
i x i = 0
j x j = 0
K x k = 0
i x j = k
j x k = i
k x i = j
j x i = -k
k x j = -i
i x k = -j
The following cross-product example can be used to form the formula for finding the vector product of two vectors in cartesian form.
a= a1i + a2j + a3k and b=b1i + b2j + b3k then,
a x b = (a1i+a2j+a3k) x (b1i+b2j+b3k)
= a1i x (b1i+b2j+b3k) + a2j x (b1i+b2j+b3k) + a3k x (b1i+b2j+b3k)
= a1i x b1i + a1i x b2j + a1i x b3k + a2j x b1i + a2j x b2j + a2j x b3k +a3k x b1i + a3k x b2j + a3k x b3k
= a1b1i x i + a1b2i x j + a1b3i x k + a2b1j x i + a2b2j x j + a2b3j x k + a3b1k x i + a3b2k x j + a3b3k x k
Now, according to the summarization we did above, three of these terms are zero. Therefore,
a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k is the cross product of two vectors formula that we can use to calculate a vector product in cartesian components of two vectors.
FAQs on Calculate Vector Product in 3D Geometry
1. What is the vector product?
The vector product, also called the cross product, is an operation between two vectors that produces a third vector perpendicular to both. It is defined for vectors in three-dimensional space and is written as π Γ π. The magnitude of the vector product is |π||π| sinΞΈ, where ΞΈ is the angle between the vectors, and its direction is given by the right-hand rule.
2. What is the formula for calculating the vector product?
The formula for the vector product of π = (aβ, aβ, aβ) and π = (bβ, bβ, bβ) is given by a determinant.
- π Γ π = | iβjβk ; aβ aβ aβ ; bβ bβ bβ |
- = i(aβbβ β aβbβ) β j(aβbβ β aβbβ) + k(aβbβ β aβbβ)
3. How do you calculate the vector product step by step?
To calculate the vector product, use the determinant method and expand carefully.
- Write vectors π and π in component form.
- Set up the determinant with i, j, k in the first row.
- Expand using cofactor expansion.
- Simplify each component.
- π Γ π = (β3, 6, β3)
4. What is the magnitude of the vector product?
The magnitude of the vector product is |π Γ π| = |π||π| sinΞΈ. Here, |π| and |π| are the magnitudes of the vectors, and ΞΈ is the angle between them. This value represents the area of the parallelogram formed by the two vectors.
5. How is the direction of the vector product determined?
The direction of the vector product is determined by the right-hand rule.
- Point your fingers in the direction of vector π.
- Rotate them toward vector π.
- Your thumb points in the direction of π Γ π.
6. What is the difference between dot product and vector product?
The dot product gives a scalar, while the vector product gives a vector perpendicular to both original vectors.
- Dot product: π Β· π = |π||π| cosΞΈ
- Vector product: π Γ π = |π||π| sinΞΈ nΜ
7. What happens if two vectors are parallel in a vector product?
If two vectors are parallel, their vector product is zero. Since the formula is |π Γ π| = |π||π| sinΞΈ, and ΞΈ = 0Β° or 180Β° for parallel vectors, sinΞΈ = 0. Therefore, π Γ π = 0.
8. Can you give an example of calculating a vector product?
Yes, here is a simple example of a vector product calculation. Let π = (2,0,1) and π = (1,3,4).
- π Γ π = | iβjβk ; 2 0 1 ; 1 3 4 |
- = i(0Β·4 β 1Β·3) β j(2Β·4 β 1Β·1) + k(2Β·3 β 0Β·1)
- = i(β3) β j(7) + k(6)
- = (β3, β7, 6)
9. What are the main properties of the vector product?
The vector product has several important algebraic properties.
- π Γ π = β(π Γ π) (anti-commutative)
- π Γ (π + π) = π Γ π + π Γ π (distributive)
- π Γ π = 0
- Result is perpendicular to both vectors
10. What is the geometric meaning of the vector product?
The geometric meaning of the vector product is that it represents a vector perpendicular to two given vectors with magnitude equal to the area of the parallelogram they form. Specifically, |π Γ π| equals the parallelogramβs area, and half of this value gives the area of the triangle formed by the vectors.
