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
1. What is a vector product, also known as a cross product?
The vector product, or cross product, of two vectors (say, a and b) is a third vector that is perpendicular to the plane containing the original two vectors. Its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them. The direction is determined by the Right-Hand Screw Rule.
2. How do you calculate the vector product of two vectors in their component form?
To calculate the vector product of two vectors, a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the most common method is using a determinant. The formula is expressed as the determinant of a 3x3 matrix:
a × b = |i j k|
|a₁ a₂ a₃|
|b₁ b₂ b₃|
Expanding this determinant gives the resultant vector: (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k.
3. What are the main algebraic properties of the vector product?
The vector product has several key properties that are crucial for calculations as per the CBSE Class 12 syllabus:
- Not Commutative: The order of multiplication matters. a × b = - (b × a). Reversing the order results in a vector with the same magnitude but opposite direction.
- Distributive over Addition: The vector product is distributive. a × (b + c) = (a × b) + (a × c).
- Product with a Scalar: For any scalar 'm', m(a × b) = (ma) × b = a × (mb).
4. How does the vector product (cross product) fundamentally differ from the scalar product (dot product)?
The fundamental difference lies in their results and geometric meaning. The scalar product (a · b) results in a single number (a scalar) that represents the projection of one vector onto another. In contrast, the vector product (a × b) results in a new vector that is perpendicular to the plane formed by the original two vectors. While the dot product answers 'how much of one vector goes along another', the cross product defines a new direction in space.
5. What is the geometric significance of the vector product of two vectors?
The geometric significance of the vector product is profound. The magnitude of the vector product, |a × b|, is equal to the area of the parallelogram that has the vectors a and b as its adjacent sides. Consequently, half of this magnitude, ½ |a × b|, represents the area of the triangle with sides a and b.
6. Why is the vector product of two parallel or collinear vectors always a zero vector?
The vector product of two parallel or collinear vectors is zero because the angle (θ) between them is either 0° (for parallel) or 180° (for anti-parallel). The formula for the magnitude of the vector product is |a||b|sin(θ). Since sin(0°) = 0 and sin(180°) = 0, the magnitude of the resultant vector is always zero. A vector with zero magnitude is a zero vector.
7. Where is the concept of the vector product used in real-world physics and engineering?
The vector product is essential in many areas of physics and engineering. For example:
- In mechanics, torque (τ) is defined as the vector product of the position vector (r) and the force vector (F), as in τ = r × F.
- Angular momentum (L) is calculated as the vector product of the position vector (r) and linear momentum (p): L = r × p.
- In electromagnetism, the magnetic force on a moving charge (the Lorentz force) is described using the vector product: F = q(v × B).
