Multiplication Of Vectors: Definition, Examples, Formula, Calculator

Multiplication Of Vectors

Edited By Vishal kumar | Updated on Jul 02, 2025 05:44 PM IST

Let us study the two-way multiplication of vectors, including the working rule, properties, applications, and examples of this type of multiplication. which belongs to the chapter Kinematics, which is an important chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), more than ten questions have been asked on this concept. And for NEET one questions were asked from this concept.

This Story also Contains

Types of Vector
Solved Example Based on Multiplication of Vector
Summary

Types of Vector

Let's start with the types of vectors.

Unit Vector

A vector having a magnitude of one unit is called a unit vector. It is represented by a cap/hat over the letter.

$\mathrm{Eg}-\hat{R}$ is called a unit vector of $\vec{R}$. Its direction is along the $\vec{R}$ and magnitude is unit.

Unit vector along $\vec{R}$ -
$
\hat{R}=\frac{\vec{R}}{|\vec{R}|}
$

Orthogonal Unit Vectors

It is defined as the unit vectors described under the three-dimensional coordinate system along the x, y, and z-axis. The three unit vectors are denoted by i, j and k respectively.

Any vector (Let us say $\vec{R}$ ) can be written as
$
\vec{R}=x \hat{i}+y \hat{j}+z \hat{k}
$

$\text { Where } \mathrm{x}, \mathrm{y} \text { and } \mathrm{z} \text { are components of } \vec{R} \text { along } \mathrm{x}, \mathrm{y} \text { and } \mathrm{z} \text { direction respectively. }$

$|\vec{R}|=\sqrt{x^2+y^2+z^2}$

Unit vector

$\hat{R}=\frac{x \hat{i}+y \hat{j}+z \hat{k}}{\sqrt{x^2+y^2+z^2}}$

If a vector is multiplied by any scalar

$\vec{Z}=n \cdot \vec{Y}$

(n=1,2,3..)

$\text { Vector } \times \text { Scalar }=\text { Vector }$

We get again a vector.

If a vector is multiplied by any real number (eg 2 or -2) then again, we get a vector quantity.

E.g.

$\text { If } \vec{A}$ is multiplied by 2 then the direction of the resultant vector is the same as that of the given vector.

$\text { Vector }=2 \vec{A}$

$\text { If } \vec{A}$ is multiplied by (-2), then the direction of the resultant is opposite to that of the given vector.

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$\text { Vector }=-2 \vec{A}$

Scalar or Dot or Inner Product

Scalar product of two vectors $\vec{A} \& \vec{B} \text { written as } \vec{A} \cdot \vec{B}$
$\vec{A} \cdot \vec{B}$ is a scalar quantity given by the product of the magnitude of $\vec{A} \& \vec{B}$ and the cosine of a smaller angle between them.

$\vec{A} \cdot \vec{B}=A B \cdot \cos \Theta $

Figure showing the representation of scalar products of vectors.

Important results-

$\begin{aligned}
& \hat{i} \cdot \hat{j}=\hat{j} \cdot \hat{k}=\hat{k} \cdot \hat{i}=0 \\
& \hat{i} \cdot \hat{i}=\hat{j} \cdot \hat{j}=\hat{k} \cdot \hat{k}=1 \\
& \vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}
\end{aligned}$

Vector or Cross-Product

Vector or cross product of two vectors $ \vec{A} \& \vec{B} \text { written as } \vec{A} \times \vec{B}$
$A \times B$ is a single vector whose magnitude is equal to the product of the magnitude of $\vec{A} \& \vec{B}$ and the sine of the smaller angle $\theta$ between them.
$\vec{A} \times \vec{B}=A B \sin \theta$

The figure shows the representation of the cross-product of vectors.

Important results

$ \begin{aligned}
& \quad \hat{i} \times \hat{j}=\hat{k}, \hat{j} \times \hat{k}=\hat{i}, \hat{k} \times \hat{i}=\hat{j} \\
& \hat{i} \times \hat{i}=\hat{j} \times \hat{j}=\hat{k} \times \hat{k}=\overrightarrow{0} \\
& \vec{A} \times \vec{B}=-\vec{B} \times \vec{A}
\end{aligned}$

Solved Example Based on Multiplication of Vector

Example 1: When we multiply a vector $\vec{A}=2 \hat{i}+3 \hat{j}-5 \hat{k}$ with a number -2, we get

1) A scalar with a magnitude same as that of the vector

2) A vector with the same magnitude and opposite direction.

3) A vector with double magnitude and opposite direction.

4) A vector with double magnitude and the same direction

Solution:

$\text { If } \vec{A}$ is multiplied by (-2), then the direction of the resultant is opposite to that of the given vector.

$\begin{aligned}
& \text { Vector }=-2 \vec{A} \\
& \vec{B}=-2 \cdot \vec{A}=2(-\vec{A})
\end{aligned}$

$\therefore$ The direction of $\vec{B} \text { is opposite of } \vec{A}$ and

Magnitude of $ |\vec{B}|=2 \times \text { magnitude of }|\vec{A}| $.

Hence, the answer is the option (3).

Example 2: Let $\vec{A}=(\hat{i}+\hat{j})$ and, $\vec{B}=(2 \hat{i}-\hat{j})$ The magnitude of a coplanar vector $\vec{C}$ such that $\vec{A} \cdot \vec{C}=\vec{B} \cdot \vec{C}=\vec{A} \cdot \vec{B}$ is given by:

1) $\sqrt{(10 / 9)}$
2) $\sqrt{5 / 9}$
3) $\sqrt{(20 / 9)}$
4) $\sqrt{(9 / 12)}$

Solution:

$\begin{aligned}
& \text { Let } \vec{c}=x \hat{i}+y \hat{j} \\
& \begin{array}{l}
\vec{A} \cdot \vec{C}=x+y \\
\vec{B} \cdot \vec{C}=2 x-y \\
\vec{A} \cdot \vec{B}=2-1=1 \\
\Rightarrow x+y=1 \\
2 x-y=1
\end{array}
\end{aligned}$

$\begin{aligned}
& 3 x=2 \quad \text { or } \mathrm{x}=2 / 3 \\
& \mathrm{y}=1 / 3 \\
& \therefore \vec{c}=2 / 3 \hat{i}+1 / 3 \hat{j} \\
& \therefore|\vec{c}|=\sqrt{(4 / 9+1 / 9)}=\sqrt{(5 / 9)}
\end{aligned}$

Hence, the answer is the option (2).

$\text { Example 3: The angle between }(\hat{l}+\hat{j}) \text { and }(\hat{l}-\hat{j}) \text { is (in degrees) }$

1) 90

2) 60

3) 45

4) 30

Solution:

As we learned

Scalar, Dot or Inner Product -

The scalar product of two vectors $\vec{A} \& \vec{B}$ written as $\vec{A} \cdot \vec{B}$ is a scalar quantity given by the product of the magnitude of $\vec{A} \& \vec{B}$ and the cosine of the smaller angle between them.

$\vec{A} \cdot \vec{B}=A B \cdot \cos \Theta$

- wherein

showing a representation of scalar products of vectors.

$\begin{aligned}
& (\hat{l}+\hat{j}) \cdot(\hat{l}-\hat{j})=|\hat{l}+\hat{j}||\hat{l}-\hat{j}| \cdot \cos \theta \\
& U \operatorname{sing} \vec{A} \cdot \vec{B}=A B \cos \theta \\
& \Rightarrow \cos \theta=\frac{(\hat{l}+\hat{j}) \cdot(\hat{l}-\hat{j})}{|\hat{l}+\hat{j}||\hat{l}-\hat{j}|}=\frac{0}{\sqrt{2} \cdot \sqrt{2}}=0 \\
& \therefore \theta=90^{\circ}
\end{aligned}$

Hence, the answer is option (1).

Example 4: A particle is thrown with $10 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ with horizontal the time at which its velocity is perpendicular to the initial velocity is ( $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}$ )

1) $\sqrt{3} \mathrm{sec}$
2) $2 \sqrt{3} \mathrm{sec}$
3) $\frac{2}{\sqrt{3}} \mathrm{sec}$
4) $\frac{4}{\sqrt{3}} \mathrm{sec}$

Solution:

$\vec{A} \cdot \vec{B}=A B \cdot \cos \Theta$

Showing the representation of scalar products of vectors.

$
\begin{aligned}
& u=10 \cos 60^{\circ} \hat{i}+10 \sin 60^{\circ} \hat{j} \\
& =5(\hat{i}+\sqrt{3} \hat{j}) \\
& V_y=u_y-g t=5 \sqrt{3}-10 t(\text { here use } g=10) \\
& \vec{V}=5 \hat{i}+(5 \sqrt{3}-10 t) \hat{j}
\end{aligned}
$
since $\vec{V}$ is perpendicular to $\vec{u} h$ ence $\vec{u} \cdot \vec{V}=0$
$
\begin{aligned}
& \Rightarrow 5(\hat{i}+\sqrt{3} \hat{j}) \cdot 5(\hat{i}+(\sqrt{3}-2 t) \hat{j})=0 \\
& \Rightarrow 1+\sqrt{3}(\sqrt{3}-2 t)=0
\end{aligned}
$
or $1+3-2 \sqrt{3} t=0$
$
\Rightarrow t=\frac{2}{\sqrt{3}} \sec
$

Hence, the answer is option (3).

Example 5: If $\vec{a}, \vec{b}$ are unit vectors such that the angle between $\vec{a}$ and $\vec{b}$ is -

1) 0
2) $\frac{\pi}{2}$
3) $\pi$
4) Indeterminate

Solution:

$\vec{A} \times \vec{B}=A B \sin \theta$

The figure shows the representation of vectors or cross product of vectors.

shows the representation of vector or cross product of vectors

$\begin{aligned}
& 0=(\vec{a}+\vec{b}) \cdot(2 \vec{a}+3 \vec{b}) \times(3 \vec{a}-2 \vec{b})=0 \\
& 0=(\vec{a}+\vec{b}) \cdot(-4 \vec{a} \times \vec{b}-9 \vec{a} \times \vec{b})=-13 \cdot(\vec{a}+\vec{b}) \cdot(\vec{a} \times \vec{b})
\end{aligned}$

which is true for all values of $\vec{a} \text { and } \vec{b}$

Summary

Vector multiplication can be divided into two categories. A vector has both a magnitude and a direction, and as a result, the dot product of two vectors and the cross product of two vectors are the two methods of multiplying vectors. Because the resultant value is a scalar quantity, the dot product of two vectors is often referred to as the scalar product of two vectors. A vector has both a magnitude and a direction associated with it.

Frequently Asked Questions (FAQs)

1. How is the dot product calculated?

The dot product of two vectors A and B is calculated as A · B = |A| |B| cos θ, where |A| and |B| are the magnitudes of vectors A and B, respectively, and θ is the angle between them. Alternatively, it can be calculated by summing the products of corresponding components: A · B = AxBx + AyBy + AzBz.

2. Can the dot product of two vectors be negative?

Yes, the dot product can be negative. This occurs when the angle between the two vectors is greater than 90° and less than 270°. A negative dot product indicates that the vectors are pointing in generally opposite directions.

3. How is the cross product calculated?

The cross product of two vectors A and B is calculated as |A × B| = |A| |B| sin θ n, where |A| and |B| are the magnitudes of vectors A and B, θ is the angle between them, and n is a unit vector perpendicular to both A and B. The direction of n is determined by the right-hand rule.

4. What does the dot product represent physically?

The dot product represents the amount of one vector that is parallel to another vector. It can be interpreted as the projection of one vector onto another, multiplied by the magnitude of the other vector.

5. What is the cross product of two vectors?

The cross product, also known as the vector product, is an operation that takes two vectors and produces a new vector that is perpendicular to both of the original vectors. The magnitude of the resulting vector is equal to the area of the parallelogram formed by the two original vectors.

6. What is vector multiplication?

Vector multiplication is an operation performed on two or more vectors to produce a new vector or scalar quantity. There are two main types of vector multiplication: dot product (scalar product) and cross product (vector product).

7. How does vector multiplication differ from scalar multiplication?

Vector multiplication involves operations between two or more vectors, while scalar multiplication involves multiplying a vector by a scalar (a single number). Vector multiplication can result in either a scalar (dot product) or a vector (cross product), whereas scalar multiplication always results in a vector.

8. What is the triple product in vector multiplication?

The triple product, also known as the scalar triple product or box product, is the dot product of one vector with the cross product of two other vectors: A · (B × C). It represents the volume of the parallelepiped formed by the three vectors.

9. What is the physical interpretation of the cross product?

The cross product represents the area of the parallelogram formed by two vectors. It also has applications in physics, such as calculating torque (the cross product of force and lever arm) or angular momentum (the cross product of position and linear momentum).

10. What is the right-hand rule in vector multiplication?

The right-hand rule is a method used to determine the direction of the resulting vector in a cross product. Point your right hand's fingers in the direction of the first vector, then curl them towards the second vector. Your thumb will point in the direction of the resulting cross product vector.

11. What is the dot product of two vectors?

The dot product, also known as the scalar product, is an operation that takes two vectors and produces a scalar (single number) result. It is calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them.

12. Can you multiply more than two vectors?

Yes, you can multiply more than two vectors, but the process becomes more complex. For example, you can perform multiple dot products or cross products in sequence, or use more advanced operations like the scalar triple product or vector triple product.

13. What is the geometric interpretation of the dot product?

Geometrically, the dot product of two vectors represents the product of the magnitude of one vector and the projection of the other vector onto it. It can be visualized as the shadow cast by one vector onto the direction of the other vector, multiplied by the length of that other vector.

14. How is the cross product used in physics?

The cross product is used in various physics applications, including:

15. What is the relationship between the dot product and the law of cosines?

The dot product is closely related to the law of cosines. In fact, the law of cosines can be derived using the dot product. If we have a triangle with sides a, b, and c, and angle C opposite side c, then: c² = a² + b² - 2ab cos C. This is equivalent to the dot product of vectors a and b: a · b = ab cos C.

16. How does vector multiplication relate to vector decomposition?

Vector multiplication, particularly the dot product, is useful in vector decomposition. To find the component of a vector A along a unit vector u, we can use the dot product: A_component = (A · u)u. This process is crucial in breaking down vectors into their components along different axes.

17. How does the concept of orthogonality relate to vector multiplication?

Orthogonal vectors are perpendicular to each other. In vector multiplication:

18. What is the significance of the scalar projection in vector multiplication?

The scalar projection of vector A onto vector B is given by (A · B) / |B|. It represents the length of the shadow cast by A onto the direction of B. This concept is crucial in understanding how vectors interact and in decomposing vectors into components along specific directions.

19. What is the relationship between vector multiplication and the parallelogram law of vector addition?

While the parallelogram law describes vector addition, it has a connection to vector multiplication through the cross product. The magnitude of the cross product |A × B| represents the area of the parallelogram formed by vectors A and B, which is the same parallelogram used to illustrate vector addition.

20. What is the significance of the scalar triple product in physics?

The scalar triple product, A · (B × C), has several important applications in physics:

21. How does vector multiplication relate to coordinate transformations?

Vector multiplication plays a role in coordinate transformations:

22. What is the significance of the dot product in calculating pressure in fluid dynamics?

In fluid dynamics, pressure is often calculated using the dot product. The force exerted by a fluid on a surface is given by F = PA, where P is pressure and A is the area vector. The component of this force normal to the surface is found using the dot product: F_normal = F · n, where n is the unit normal vector to the surface.

23. How does vector multiplication relate to the concept of mechanical advantage in simple machines?

Vector multiplication, particularly the

24. Is the cross product commutative?

No, the cross product is not commutative. The order of the vectors matters in cross product multiplication. A × B = -(B × A), meaning the result of B × A is equal in magnitude but opposite in direction to A × B.

25. How does the magnitude of the cross product relate to the angle between vectors?

The magnitude of the cross product is maximum when the vectors are perpendicular (90°) to each other and zero when they are parallel or antiparallel (0° or 180°). This is because |A × B| = |A| |B| sin θ, and sin 90° = 1, while sin 0° = sin 180° = 0.

26. Is the triple product (A · (B × C)) the same as ((A × B) · C)?

Yes, the triple product is the same regardless of which two vectors are crossed first. This is known as the triple product identity: A · (B × C) = B · (C × A) = C · (A × B).

27. What is the difference between scalar triple product and vector triple product?

The scalar triple product (A · (B × C)) results in a scalar value, while the vector triple product (A × (B × C)) results in a vector. The vector triple product can be expanded using the vector triple product identity: A × (B × C) = B(A · C) - C(A · B).

28. How does vector multiplication apply to motion in a straight line?

While motion in a straight line primarily involves scalar quantities, vector multiplication becomes important when considering forces acting on an object, even if the motion is linear. For example, the dot product can be used to find the component of a force acting along the direction of motion.

29. How does the dot product relate to the angle between vectors?

The dot product is directly related to the cosine of the angle between two vectors. If A · B = |A| |B| cos θ, then cos θ = (A · B) / (|A| |B|). This relationship allows us to find the angle between two vectors using their dot product and magnitudes.

30. What happens when you take the dot product of a vector with itself?

When you take the dot product of a vector with itself, the result is equal to the square of the vector's magnitude. This is because A · A = |A| |A| cos 0° = |A|², as the angle between a vector and itself is 0°, and cos 0° = 1.

31. How does the distributive property apply to vector multiplication?

The distributive property applies to both dot and cross products:

32. What is the significance of perpendicular vectors in vector multiplication?

Perpendicular vectors (vectors at right angles to each other) have special properties in vector multiplication:

33. What is the difference between parallel and antiparallel vectors in terms of their dot and cross products?

For parallel vectors (pointing in the same direction):

34. How does the associative property apply to vector multiplication?

The associative property does not apply to vector multiplication in the same way it does to scalar multiplication. For dot products, (A · B) · C is not defined because A · B is a scalar, not a vector. For cross products, A × (B × C) ≠ (A × B) × C. However, the scalar triple product is associative: A · (B × C) = (A · B) × C.

35. What is the significance of unit vectors in vector multiplication?

Unit vectors (vectors with a magnitude of 1) are important in vector multiplication because:

36. How does vector multiplication relate to the concept of work in physics?

In physics, work is defined as the dot product of force and displacement vectors: W = F · d. This multiplication captures both the magnitude of the force and displacement, as well as the angle between them, providing a scalar quantity that represents the energy transferred by the force.

37. What is the relationship between vector multiplication and vector addition?

While vector addition is a simpler operation that combines vectors, vector multiplication (both dot and cross products) provides information about the relationships between vectors, such as their relative orientations and magnitudes. Vector multiplication is not used to combine vectors in the same way as vector addition.

38. How does vector multiplication apply to rotational motion?

Vector multiplication, particularly the cross product, is fundamental in describing rotational motion:

39. What is the significance of the vector triple product expansion?

The vector triple product expansion, A × (B × C) = B(A · C) - C(A · B), is a powerful tool in vector algebra. It allows us to simplify complex cross product expressions into combinations of dot products, which are often easier to calculate. This expansion is particularly useful in electromagnetic theory and fluid dynamics.

40. How does vector multiplication relate to the concept of moments in physics?

In physics, moments (such as torque) are often calculated using the cross product. For example, the moment of a force F about a point O is given by M = r × F, where r is the position vector from O to the point of application of the force. This cross product captures both the magnitude of the moment and its axis of rotation.

41. How does vector multiplication apply to electromagnetic theory?

Vector multiplication is crucial in electromagnetic theory:

42. What is the relationship between vector multiplication and vector fields?

Vector multiplication is fundamental in describing and analyzing vector fields:

43. How does vector multiplication apply to the concept of power in physics?

In physics, power is often calculated using the dot product. For example, the power delivered by a force F acting on an object moving with velocity v is given by P = F · v. This dot product captures both the magnitude of the force and velocity, as well as the angle between them, providing a scalar quantity representing the rate of energy transfer.

44. What is the significance of the vector product in calculating angular momentum?

The angular momentum L of a particle is defined as the cross product of its position vector r and its linear momentum p: L = r × p. This cross product captures both the magnitude of the angular momentum and its axis of rotation, which is perpendicular to both r and p.

45. How does vector multiplication relate to the concept of flux in physics?

Flux, which represents the flow of a quantity through a surface, often involves vector multiplication:

46. What is the relationship between vector multiplication and the concept of work-energy theorem?

The work-energy theorem states that the work done on an object equals its change in kinetic energy. This theorem relies on the dot product to calculate work: W = F · d = ΔKE. The dot product ensures that only the component of force parallel to the displacement contributes to the work done.

47. How does vector multiplication apply to the concept of torque in rotational dynamics?

Torque, which causes rotational acceleration, is calculated using the cross product: τ = r × F, where r is the position vector from the axis of rotation to the point of application of force F. This cross product captures both the magnitude of the torque and its axis of rotation, which is perpendicular to both r and F.