Product of Vectors

Product of vectors is of two types. A vector has both magnitude and direction and based on this the two product of vectors are, the dot product of two vectors and the cross product of two vectors. The dot product of two vectors is also referred to as scalar product, as the resultant value is a scalar quantity. The cross product is called the vector product as the result is a vector, which is perpendicular to these two vectors.

Let us learn about the two product of vectors, the working rules, properties, uses, examples of these product of vectors.

A vector has both magnitude and direction. We can multiply two or more vectors by dot product and cross product. Let us understand more about each of the products of vectors.

Dot Product

The dot product of vectors is also called the scalar product of vectors. The resultant of the dot product of the vectors is a scalar value. Dot Product of vectors is equal to the product of the magnitudes of the two vectors, and the cosine of the angle between the two vectors. The resultant of the dot product of two vectors lie in the same plane of the two vectors. The dot product may be a positive real number or a negative real number.

Let a and b be two non-zero vectors, and θ be the included angle of the vectors. Then the scalar product or dot product is denoted by a.b, which is defined as:

\(\overrightarrow a. \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos θ.

Here, \(|\overrightarrow a|\) is the magnitude of \(\overrightarrow a\), \(|\overrightarrow b|\) is the magnitude of \(\overrightarrow b\), and θ is the angle between them.

Cross Product

Cross Product is also called a Vector Product. Cross product is a form of vector multiplication, performed between two vectors of different nature or kinds. When two vectors are multiplied with each other and the product 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.

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:

\(\overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{c}\)

Here \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are two vectors, and \(\overrightarrow{c}\) is the resultant vector. 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:

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

The working rule for the product of two vectors, the dot product, and the cross product can be understood from the below sentences.

Dot Product

For the dot product of two vectors, the two vectors are expressed in terms of unit vectors, i, j, k, along the x, y, z axes, then the scalar product is obtained as follows:

If \(\overrightarrow a = a_1\hat i + b_1 \hat j + c_1 \hat k\) and \(\overrightarrow b = a_2 \hat i + b_2 \hat j + c_2\hat k\), then

\(\overrightarrow a. \overrightarrow b\) = \((a_1 \hat i + b_1 \hat j + c_1 \hat k)(a_2 \hat i + b_2 \hat j + c_2 \hat k)\)

= \((a_1a_2) (\hat i. \hat i) + (a_1b_2) (\hat i.\hat j)+ (a_1c_2) (\hat i. \hat k) + \\(b_1a_2) (\hat j. \hat i) + (b_1b_2)(\hat j. \hat j) + (b_1c_2 (\hat j. \hat k) + \\(c_1a_2)(\hat k. \hat i) + (c_1b_2)(\hat k. \hat j) + (c_1c_2)(\hat k. \hat k)\)

\(\overrightarrow a. \overrightarrow b\) = \(a_1a_2 + b_1b_2+ c_1c_2\)

Cross Product

Let us assume that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are two vectors, such that \(\overrightarrow{a}\)= \(a_1\hat i+b_1 \hat j+c_1 \hat k\) and \(\overrightarrow{b}\) = \(a_2 \hat i+b_2 \hat j+c_2 \hat k\) then by using determinants, we could find the cross product and write the result as the cross product formula using the following matrix notation.

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

\(\overrightarrow{a} \times \overrightarrow{b} = \hat i (b_1c_2-b_2c_1) - \hat j (a_1c_2-a_2c_1) + \hat k (a_1b_2-a_2b_1)\)

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.

The dot product of the unit vector is studied by taking the unit vectors \(\hat i\) along the x-axis, \(\hat j\) along the y-axis, and \(\hat k\) along the z-axis respectively. The dot product of unit vectors \(\hat i\), \(\hat j\), \(\hat k\) follows similar rules as the dot product of vectors. The angle between the same vectors is equal to 0º, and hence their dot product is equal to 1. And the angle between two perpendicular vectors is 90º, and their dot product is equal to 0.

• \(\hat i.\hat i\) = \(\hat j.\hat j\) = \(\hat k.\hat k\)= 1
• \(\hat i.\hat j\) = \(\hat j.\hat k\) = \(\hat k.\hat i\)= 0

The cross product of unit vectors \(\hat i\), \(\hat j\), \(\hat k\) follows similar rules as the cross product of vectors. The angle between the same vectors is equal to 0º, and hence their cross product is equal to 0. And the angle between two perpendicular vectors is 90º, and their cross product gives a vector, which is perpendicular to the two given vectors.

• \(\overrightarrow{i}\times \overrightarrow{i} =\overrightarrow{j}\times \overrightarrow{j} = \overrightarrow{k}\times \overrightarrow{k} = 0\)

The cross product of two vectors follow a cyclic order as in the below image. The cross product of two vector in the cyclical sequence gives the third vector in sequence.

• \(\overrightarrow{i}\times \overrightarrow{j} = \overrightarrow{k}; \overrightarrow{j}\times \overrightarrow{k}= \overrightarrow{i}; \overrightarrow{k}\times \overrightarrow{i} = \overrightarrow{j}\)
• \(\overrightarrow{j}\times \overrightarrow{i} = \overrightarrow{-k}; \overrightarrow{k}\times \overrightarrow{j}= \overrightarrow{-i}; \overrightarrow{i}\times \overrightarrow{k} = \overrightarrow{-j}\)

The properties of the product of vectors are helpful to gain a detailed understanding of vectors multiplication and also to perform numerous calculations involving vectors, A few important properties of product of vectors are listed here.

1. The cross product of two vectors is given by the formula \( \overrightarrow{a} \times \overrightarrow{b} = |a| |b| \sin(\theta)\).
2. The dot product of two vectors is given by the formula \( \overrightarrow{a} . \overrightarrow{b} = |a| |b| \cos(\theta)\).
3. The dot product of two vectors follows the commutative property. \(\vec a. \vec b = \vec b. \vec a \)
4. The cross-product of two vectors do no follow the commutative property. \( \vec a \times \vec b\neq \vec b \times \vec a \)
5. Anti-commutative property: \(\overrightarrow{a} \times \overrightarrow{b} = - \overrightarrow{b} \times \overrightarrow{a}\)
6. Distributive property: \(\overrightarrow{a} \times (\overrightarrow{b} + \overrightarrow{c}) = (\overrightarrow{a}\times \overrightarrow{b} )+ (\overrightarrow{a}\times \overrightarrow{c})\)
7. Cross product of the zero vector: \(\overrightarrow{a}\times \overrightarrow{0} = \overrightarrow{0}\)
8. Cross product of the vector with itself: \(\overrightarrow{a}\times \overrightarrow{a} = \overrightarrow{0}\)
9. Multiplied by a scalar quantity:\(c(\overrightarrow{a}\times \overrightarrow{b}) = c\overrightarrow{a}\times \overrightarrow{b} = \overrightarrow{a}\times c\overrightarrow{b}\)
10. The dot product of two vectors is a scalar and lies in the plane of the two vectors.
11. The cross product of two vectors is a vector, which is perpendicular to the plane containing these two vectors.

The following are some of the important uses of the product of vectors. Let us understand about each of these uses in the below paragraphs.

• Projection of a Vector
• Angle Between Two Vectors
• Triple Cross Product
• Area of a Parallelogram
• Volume of a Parallelepiped

Projection of a Vector

The dot product is useful for finding the component of one vector in the direction of the other. The vector projection of one vector over another vector is the length of the shadow of the given vector over another vector. It is obtained by multiplying the magnitude of the given vectors with the cosecant of the angle between the two vectors. The resultant of a vector projection formula is a scalar value.

Here \(\overrightarrow a\), and \(\overrightarrow b\), are two vectors and θ is the angle between the two vectors. The projection of \(\overrightarrow a\) on \(\overrightarrow b\) is \(|\overrightarrow a|\) cos θ. This can be further simplified to obtain the following formula for the project of a vector.

Thus, projection of \(\overrightarrow a\) on \(\overrightarrow b = \dfrac{\overrightarrow a. \overrightarrow b}{|\overrightarrow b|}\)

Angle Between Two Vectors

The angle between two vectors is calculated as the cosine of the angle between the two vectors. The cosine of the angle between two vectors is equal to the sum of the product of the individual constituents of the two vectors, divided by the product of the magnitude of the two vectors. The formula for the angle between the two vectors is as follows.

\(cos\theta = \dfrac{\overrightarrow a.\overrightarrow b}{|a|.|b|}\)

\(cos\theta = \dfrac{a_1.b_1 + a_2.b_2 +a_3.b_3}{\sqrt{a_1^2 + a_2^2 +a_3^3}.\sqrt{b_1^2 + b_2^2 + b_3^2}}\)

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:

\((\overrightarrow{a}\times \overrightarrow{b}) \times \overrightarrow{c} = (\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b} -(\overrightarrow{b}\cdot \overrightarrow{c}) \overrightarrow{a}\)

Area of a Parallelogram

The two adjacent sides of a parallelogram can be represented by the vectors \(\overrightarrow a\), and \(\overrightarrow b\). The area of the parallelogram is the product of the base and the height of the parallelogram. Let us consider the base of the parallelogram as \(|\overrightarrow a|\), and the height of the parallelogram as \(|\overrightarrow b|\)sin θ.

Here Base = \(|\overrightarrow a|\), Height = \(|\overrightarrow b|\)sin θ, and the Area of the parallelogram = Base x Height

Area of the Parallelogram = \(|\overrightarrow a|.|\overrightarrow b|\)sin θ = \(\overrightarrow a \times \overrightarrow b \)

Volume of a Parallelepiped

A parallelepiped is a six-sided figure, each of whose sides is a parallelogram. Here the opposite side parallelograms are identical. The volume V of the parallelepiped can be obtained from the side of edges a, b, c. The volume of a parallelepiped can be obtained from the product of the area of the base and the height of the parallelepiped. The base area of the parallelepiped is |b x c| and the height of the parallelepiped is |a|. The formula for the calculation of the volume of a parallelepiped is as follows.

V = a.(b x c)

Related Topics

The following topics help in a better understanding of product of vectors.

• Collinear Vectors
• Vectors
• Types of Vectors
• Addition of Vectors
• Cross Product of Two Vectors

What is the Dot Product of Vectors?

The dot product of two vectors has two definitions. Algebraically the dot product of two vectors is equal to the sum of the products of the individual components of the two vectors. a.b = \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\). Geometrically the dot product of two vectors is the product of the magnitude of the vectors and the cosine of the angle between the two vectors. ( \(\overrightarrow a. \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos θ). The resultant of the dot product of vectors is a scalar value.

How To Calculate the Dot Product of Vectors?

The dot product can be calculated in three simple steps. First find the magnitude of the two vectors a and b, ie |a| and |b|. Secondly, find the cosecant of the angle θ between the two vectors. Finally take a product of the magnitude of the two vectors and the and cosecant of the angle between the two vectors, to obtain the dot product of the two vectors. (a.b = |a|.|b|.Cosθ. Also check to dot product calculator, to easily find the vector dot product.

Why is the Dot Product Called Scalar Product of Vectors?

The dot product is a scalar because all the individual constituents of the answer are scalar values. In a.b = |a|.|b|.Cosθ, |a|, |b|, and Cosθ are all scalar values. Hence the dot product is also called a scalar product.

Why Do We Use Cos in Dot Product of Vectors?

For finding the dot product we need to have the two vectors a, b in the same direction. Since the vectors, a and b are at an angle to each other, the value acosθ is the component of vector a in the direction of vector b. Hence we find cosθ in the dot product of two vectors.

What is The Cross Product of 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 Dot Product and Cross Product of 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: \(\overrightarrow{a}. \overrightarrow{b} = |a| |b| \cos(\theta)\).

What is the Difference Between Dot Product and Cross Product of 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 Right Hand Thumb Rule for Cross Product of 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.