Components of vector Along and Perpendicular to another Vector

Q: How does the angle between two vectors affect their components?

The angle between vectors directly influences the magnitudes of their components. As the angle approaches 0°, the parallel component increases and the perpendicular component decreases. As the angle approaches 90°, the opposite occurs. At 0°, the parallel component equals the vector's magnitude, and at 90°, the perpendicular component equals the vector's magnitude.

Q: How do you express a vector as the sum of its parallel and perpendicular components?

A vector v can be expressed as v = v_parallel + v_perpendicular, where v_parallel is the component parallel to the reference vector and v_perpendicular is the component perpendicular to it. This sum represents the complete decomposition of the vector.

Q: Why is it impossible for both the parallel and perpendicular components to be zero simultaneously (for a non-zero vector)?

It's impossible because the original vector is the sum of these components. If both were zero, it would imply the original vector is a zero vector. For any non-zero vector, at least one of its components must be non-zero to maintain the vector's magnitude and direction.

Q: Why is it useful to find components of a vector along and perpendicular to another vector?

Finding these components is useful in many physics and engineering applications. It allows us to analyze forces, velocities, or other vector quantities in specific directions of interest, simplifying complex problems and providing insights into how vectors interact in different directions.

Q: What's the geometric interpretation of the dot product in relation to vector components?

Geometrically, the dot product represents the product of the magnitudes of two vectors and the cosine of the angle between them. This relates directly to finding the parallel component, as it gives the projection of one vector onto another, scaled by the magnitude of the reference vector.

Q: What does it mean to find the component of a vector along another vector?

Finding the component of a vector along another vector means determining how much of the first vector points in the same direction as the second vector. It's like projecting the first vector onto the second vector, giving us the part of the first vector that aligns with the second vector's direction.

Q: Why is the dot product used to find the component of a vector along another vector?

The dot product is used because it gives the scalar projection of one vector onto another. When we divide the dot product of two vectors by the magnitude of the reference vector, we get the length of the component along that vector. This method efficiently calculates the parallel component's magnitude.

Q: Can the component of a vector along another vector be negative?

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

Q: How is the component of a vector perpendicular to another vector different from the parallel component?

The perpendicular component is the part of the vector that is at right angles to the reference vector, while the parallel component is the part that aligns with the reference vector. Together, these two components form the original vector, representing its decomposition along and perpendicular to the reference vector.

Q: What's the relationship between vector decomposition and finding components along and perpendicular to another vector?

Vector decomposition is the process of breaking a vector into its components along different directions. Finding components along and perpendicular to another vector is a specific case of vector decomposition where we choose the reference vector as one of the directions for decomposition.

Components of a Vector Along and Perpendicular to another Vector

Edited By Komal Miglani | Updated on Jul 02, 2025 07:46 PM IST

Let's first revise the vector concept before knowing about the component of the vector. A quantity that has magnitude as well as a direction in space and follows the triangle law of addition is called a vector quantity, e.g., velocity, force, displacement, etc. in real life we use vectors for tracking objects that are in motion, and localization of places and things.

This Story also Contains

Projection of vector
Scalar Projection
Solved Examples

Components of a Vector Along and Perpendicular to another Vector

In this article, we will cover the concept of Finding Components of a vector Along and Perpendicular to another Vector. This topic falls under the broader category of Vector Algebra, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE. A total of ten questions have been asked on this topic in JEE Main from 2013 to 2023 including one in 2018 and one in 2023.

Projection of vector

The vector projection is of two types: Scalar projection which tells about the magnitude of vector projection and the other is the Vector projection which says about itself and represents the unit vector.

Scalar Projection

Let a→ and b→ be two vectors then,

Projection of a→ on b→=a→⋅b→|b→|=a→⋅b→|b→|=a→⋅b^
Projection of b→ on a→=a→⋅b→|a→|=b→⋅a→|a→|=b→⋅a^

Vector Projection
Let a→ and b→ be two vectors represented by OA→ and OB→ respectively and let ⊖ be the angle between a→ and b→. Then,

b→=OM→+MB→
Also,

OM→=(OM)a^=(OBcos⁡θ)a^==(|b→|cos⁡θ)a^=((a→⋅b→)|a→|)a^=((a→⋅b→)|a→||a→|)a→[∵a^=a→|a→|]=((a→⋅b→)|a→|2)a→

As,

b→=OM→+MB→MB→=b→−OM→=b→−((a→⋅b→)|a→|2)a→
Thus, the components of b→ along and perpendicular to a→ are ((a→⋅b→)|a→|2)a→ and b→−((a→⋅b→)|a→|2)a→, respectively.

Solved Examples

Example 1: The length of the projection of the line segment joining the points (5,−1,4) and (4,−1,3) on the plane, x+y+z=7 is : [JEE MAINS 2018]
Solution

Projection of a line segment on a line - Projection of the line segment joining the points P(x1,y1,z1) and Q(x2,y2,z2) on a line having direction cosines (l,m,n) is (x2−x1)l+(y2−y1)m+(z2−z1)n

- wherein

Here PQ is the projection
Also PQ=BC

AC=AB^⋅BC^=(i^+k^)⋅(i^+j^+k^3)=2/3
Also AB=2

AB=AC2+BC2=2=4/3+BC2BC2=2/3
Example 2: Let ABCD be a parallelogram such that AB→=q→,AD→=p→ and ∠BAD be an acute angle. If r→ is the vector that coincides with the altitude directed from the vertex B to the side AD, then r→ is given by Solution:

r→=BA→+AQ→

=−q→+ projection of BA→ across AD→=−q→+(p→⋅q→)p→(p→⋅p→)

Projection of vector b on vector a -

b→cos⁡Θ=a→⋅b→|a→|

dot product

Example 3: Let u→,v→,w→ be such that |u→|=1,|v→|=2,|w→|=3. If the projection v→ along u→ is equal to that of w→ along u→ and v→,w→ are perpendicular to each other then |u→−v→+w→| equals

Solution:

Projection of vector b on vector a b→cos⁡θ=a→⋅b→|a→|

dot product

Scalar Product of two vectors -
a→⋅b→>0 an acute angle
a→⋅b→<0 an obtuse angle
a→⋅b→=0 a right angle
- wherein
Θ is the angle between the vectors a→ and b→

v→⋅u→|u→|=w→⋅u→|u→|⇒v→⋅u→=w→⋅u→
Also, ⇒v→⋅w→=0

⇒|u→v→+w→|=|u→|2+|v→|2+|w→|2=1+4+9=14

Example 4:

The magnitude of the projection of vector 2i^+3j^+k^ on the vector perpendicular to the plane containing the vectors i^+j^+k^ and i^+2j^+3k^ is

Solution

Projection of vector b on vector a -

$
\vec{b} \cos \Theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}

Normal vector to the plane containing vector i^+j^+k^
and i^+2j^+3k^ is

n^=(i^+j^+k^)×(i^+2j^+3k^)n^=i^−2j^+k^
Projection of 2i^+3j^+k^ on n^

=|(2i^+3j^+k^)(i^−2j^+k^)1+4+1|

=36=32

Example 5: The distance of the point having position vector −i^+2j^+6k^ from the straight line passing through the point ( 2, 3 , -4 ) and parallel to the vector, 6i^+3j^−4k^ is :

Solution

Projection of vector b on vector a -b→cos⁡Θ=a→⋅b→|a→|

Point P(-1,2,6) & A(2,3,-4)

and n→=6i^+3j^−4k^

x→=6i^+3j^−4k^

PD2=AP2−AD2

AD=|AP→⋅n→|n→|

AD=|(3i^+j^−10k^)⋅(6i^+3j^−4k^)|6i^+3j^−4k^|=61PD=110−61=7

Summary

Understanding vector components is fundamental for analyzing and manipulating vectors. Components provide a clear and systematic way to express the influence of a vector along specific axes or directions within a coordinate system. This concept is essential for solving equations, performing calculations, and interpreting physical phenomena accurately

Frequently Asked Questions (FAQs)

1. How does the angle between two vectors affect their components?

The angle between vectors directly influences the magnitudes of their components. As the angle approaches 0°, the parallel component increases and the perpendicular component decreases. As the angle approaches 90°, the opposite occurs. At 0°, the parallel component equals the vector's magnitude, and at 90°, the perpendicular component equals the vector's magnitude.

2. How do you express a vector as the sum of its parallel and perpendicular components?

A vector v can be expressed as v = v_parallel + v_perpendicular, where v_parallel is the component parallel to the reference vector and v_perpendicular is the component perpendicular to it. This sum represents the complete decomposition of the vector.

3. Why is it impossible for both the parallel and perpendicular components to be zero simultaneously (for a non-zero vector)?

It's impossible because the original vector is the sum of these components. If both were zero, it would imply the original vector is a zero vector. For any non-zero vector, at least one of its components must be non-zero to maintain the vector's magnitude and direction.

4. Why is it useful to find components of a vector along and perpendicular to another vector?

Finding these components is useful in many physics and engineering applications. It allows us to analyze forces, velocities, or other vector quantities in specific directions of interest, simplifying complex problems and providing insights into how vectors interact in different directions.

5. What's the geometric interpretation of the dot product in relation to vector components?

Geometrically, the dot product represents the product of the magnitudes of two vectors and the cosine of the angle between them. This relates directly to finding the parallel component, as it gives the projection of one vector onto another, scaled by the magnitude of the reference vector.

6. What does it mean to find the component of a vector along another vector?

Finding the component of a vector along another vector means determining how much of the first vector points in the same direction as the second vector. It's like projecting the first vector onto the second vector, giving us the part of the first vector that aligns with the second vector's direction.

7. Why is the dot product used to find the component of a vector along another vector?

The dot product is used because it gives the scalar projection of one vector onto another. When we divide the dot product of two vectors by the magnitude of the reference vector, we get the length of the component along that vector. This method efficiently calculates the parallel component's magnitude.

8. Can the component of a vector along another vector be negative?

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

9. How is the component of a vector perpendicular to another vector different from the parallel component?

The perpendicular component is the part of the vector that is at right angles to the reference vector, while the parallel component is the part that aligns with the reference vector. Together, these two components form the original vector, representing its decomposition along and perpendicular to the reference vector.

10. What's the relationship between vector decomposition and finding components along and perpendicular to another vector?

Vector decomposition is the process of breaking a vector into its components along different directions. Finding components along and perpendicular to another vector is a specific case of vector decomposition where we choose the reference vector as one of the directions for decomposition.

11. How do you calculate the magnitude of the perpendicular component once you know the parallel component?

To find the magnitude of the perpendicular component, use the Pythagorean theorem. If |v| is the magnitude of the original vector and |v_parallel| is the magnitude of the parallel component, then |v_perpendicular| = √(|v|² - |v_parallel|²).

12. How do unit vectors simplify the process of finding vector components?

Unit vectors simplify the process because they have a magnitude of 1. When finding components, using unit vectors eliminates the need to divide by the magnitude of the reference vector. The dot product of a vector with a unit vector directly gives the component's magnitude in that direction.

13. How do you find the angle between two vectors using their components?

To find the angle between two vectors, use the dot product formula: a · b = |a||b|cos(θ). Rearrange this to solve for θ: θ = arccos((a · b) / (|a||b|)). This method uses the components of vectors to calculate their dot product and magnitudes.

14. Can you have negative perpendicular components? What does this mean geometrically?

The magnitude of a perpendicular component is always positive, but we can assign a negative sign to indicate direction in a 2D plane. Geometrically, a "negative" perpendicular component would point in the opposite direction to what we've defined as positive, but it's still perpendicular to the reference vector.

15. How do vector components change if you reverse the direction of the reference vector?

If you reverse the direction of the reference vector, the parallel component changes sign (positive becomes negative and vice versa), while the perpendicular component remains unchanged. This is because the parallel component depends on the direction of the reference vector, but perpendicularity is maintained regardless of the reference vector's direction.

16. How does the process of finding components change when dealing with non-orthogonal coordinate systems?

In non-orthogonal systems, the process becomes more complex. We can't simply use dot products with unit vectors along the axes. Instead, we need to use the metric tensor or reciprocal basis vectors to find components. The concept remains similar, but the calculations involve more steps and consideration of the system's geometry.

17. What's the relationship between vector components and vector addition?

Vector components are fundamentally related to vector addition. When we decompose a vector into its components, we're essentially finding two (or more) vectors that, when added together, give us the original vector. This principle is key in many physics and engineering applications where we need to combine or analyze forces or other vector quantities.

18. Can you have more than two components for a vector? If so, when and why?

Yes, you can have more than two components, especially in higher-dimensional spaces or when using multiple reference vectors. For example, in 3D space, we often use three components along the x, y, and z axes. Multiple components are useful when analyzing vector behavior in complex systems or when working in spaces with more than three dimensions.

19. How do vector components relate to the concept of vector spaces?

Vector components are fundamental to understanding vector spaces. In a vector space, any vector can be expressed as a linear combination of basis vectors. When we find components, we're essentially expressing a vector in terms of a particular basis (like the standard basis in Cartesian coordinates, or a basis defined by reference vectors).

20. What role do vector components play in understanding and solving systems of linear equations?

Vector components are crucial in systems of linear equations. Each equation in the system can be thought of as defining a hyperplane in the vector space. The solution to the system is the point (or set of points) where all these hyperplanes intersect. Understanding how vectors decompose into components helps in visualizing and solving these systems.

21. How does the concept of vector components extend to quantum mechanics and other advanced physics?

In quantum mechanics, vector components extend to complex vector spaces. State vectors in quantum systems are often expressed as components in various bases (e.g., position or momentum basis). Understanding how to decompose vectors and change bases is crucial for calculating probabilities and expectation values of observables in quantum systems.

22. What's the connection between vector components and the gradient in multivariable calculus?

The gradient of a scalar field is a vector whose components are the partial derivatives with respect to each variable. Each component of the gradient represents the rate of change of the function in that direction. Understanding vector components is crucial for interpreting and working with gradients in optimization problems and physical applications.

23. What's the significance of vector components in electrical engineering, particularly in AC circuit analysis?

In AC circuit analysis, voltages and currents are often represented as complex vectors. The real and imaginary components of these vectors correspond to the in-phase and quadrature components of the AC signals. Understanding vector components in this context helps engineers analyze phase relationships, power factors, and energy transfer in AC systems.

24. How do vector components relate to the concept of cross products?

While dot products relate directly to parallel components, cross products are closely tied to perpendicular components. The magnitude of the cross product of two vectors is equal to the product of the magnitude of one vector and the magnitude of the perpendicular component of the other vector. This relationship is fundamental in understanding torque, angular momentum, and magnetic force calculations.

25. What role do vector components play in the theory of special relativity?

In special relativity, four-vectors represent events in spacetime. Understanding how these vectors decompose into time and space components in different reference frames is crucial for grasping concepts like time dilation and length contraction. Lorentz transformations, which relate observations in different inertial frames, involve manipulating these vector components.

26. How are vector components used in optimization problems and machine learning algorithms?

In optimization and machine learning, vectors often represent parameters or gradients. Understanding how these vectors decompose into components is crucial for implementing algorithms like gradient descent. In high-dimensional spaces, analyzing and manipulating vector components is key to finding optimal solutions and understanding the behavior of complex models.

27. Can you explain how vector components are used in crystallography and materials science?

In crystallography, vector components are used to describe crystal structures and lattice directions. Miller indices, which specify planes in crystal lattices, are essentially reciprocal vector components. Understanding how to decompose and analyze vectors in crystal coordinate systems is crucial for studying material properties, X-ray diffraction patterns, and crystal symmetries.

28. How do vector components relate to the concept of tensors in physics and engineering?

Tensors are generalizations of vectors that can have multiple indices. Vector components can be seen as the simplest non-trivial tensors (rank-1 tensors). Understanding how vectors decompose into components is a stepping stone to grasping more complex tensor operations, which are crucial in areas like continuum mechanics, general relativity, and electromagnetic theory.

29. How are vector components used in computer vision and image processing?

In computer vision, vectors often represent features, gradients, or optical flow. Understanding how to decompose and analyze these vectors in different coordinate systems is crucial for tasks like edge detection, feature matching, and motion tracking. Vector components also play a key role in transformations like rotation and scaling of images.

30. Can you explain how vector components are used in quantum field theory?

In quantum field theory, fields are often represented as vectors in abstract spaces. Understanding how these vectors decompose into components in different bases is crucial for calculating scattering amplitudes and understanding particle interactions. The concept of vector components extends to more complex objects like spinors and tensors in advanced quantum field theories.

31. What's the role of vector components in understanding and calculating stress and strain in materials science?

In materials science, stress and strain are represented by tensors, which can be thought of as collections of vector components. Understanding how these components relate to different directions in the material is crucial for analyzing material behavior under various loading conditions. Vector components are essential in calculating principal stresses and strains, and in understanding concepts like yield criteria.

32. How are vector components used in robotics, particularly in inverse kinematics problems?

In robotics, vector components are crucial for

33. Can the perpendicular component of a vector be zero? If so, what does this mean geometrically?

Yes, the perpendicular component can be zero. Geometrically, this means the two vectors are parallel or anti-parallel (pointing in the same or exactly opposite directions). In this case, the original vector lies entirely along the direction of the reference vector.

34. What happens to the components when the reference vector is scaled?

Scaling the reference vector doesn't change the direction of the components, but it does affect their magnitudes. The parallel component's magnitude changes inversely with the scaling of the reference vector, while the perpendicular component remains unchanged in magnitude.

35. How does the concept of vector components relate to vector projection?

Vector projection is essentially finding the parallel component of a vector. When we project vector a onto vector b, we're finding the component of a that lies along b. The perpendicular component is what remains after subtracting the projection from the original vector.

36. What's the relationship between vector components and vector resolution in physics problems?

Vector resolution in physics often involves finding components along specific directions, typically horizontal and vertical. Finding components along and perpendicular to another vector is a more general case of this process, allowing resolution along any chosen direction, not just standard coordinate axes.

37. How do you find the component of a 3D vector along another 3D vector?

For 3D vectors, use the same dot product method: v_parallel = (v · u) u / |u|², where v is the vector being decomposed, u is the reference vector, and |u| is the magnitude of u. This gives the parallel component as a vector. The perpendicular component can then be found by v_perpendicular = v - v_parallel.

38. What's the significance of the scalar projection in finding vector components?

The scalar projection is the magnitude of the parallel component. It's significant because it tells us how much of the original vector's length is in the direction of the reference vector. Multiplying this scalar by the unit vector in the reference direction gives us the parallel component vector.

39. Can you explain how vector components relate to the concept of vector basis?

Vector components are essentially the coefficients of a vector when expressed in terms of a particular basis. When we find components along and perpendicular to another vector, we're effectively creating a new two-vector basis (the reference vector and its perpendicular) and expressing our original vector in terms of this basis.

40. How do you verify that the parallel and perpendicular components you've calculated are correct?

To verify, you can: 1) Check that the sum of the parallel and perpendicular components equals the original vector. 2) Ensure the dot product of the perpendicular component with the reference vector is zero. 3) Confirm that the magnitudes satisfy the Pythagorean theorem: |v|² = |v_parallel|² + |v_perpendicular|².

41. How does the concept of orthogonality apply to vector components?

Orthogonality is crucial in vector components. The parallel and perpendicular components are, by definition, orthogonal (perpendicular) to each other. This orthogonality ensures that these components represent independent aspects of the original vector, allowing us to analyze its behavior in different directions separately.

42. How do vector components relate to the concept of work in physics?

In physics, work is defined as the dot product of force and displacement vectors. This can be interpreted as finding the component of the force vector along the direction of displacement. Understanding vector components helps in calculating work in various scenarios, especially when forces are not aligned with the direction of motion.

43. Can you explain how vector components are used in computer graphics and 3D modeling?

In computer graphics and 3D modeling, vector components are essential for representing and manipulating objects in space. Points and directions are typically represented as vectors with x, y, and z components. Transformations like rotation and scaling involve manipulating these components. Understanding how to decompose and recombine vectors is crucial for implementing realistic 3D graphics and animations.

44. How do vector components help in understanding and calculating moments and torques in physics and engineering?

Moments and torques are vector quantities that depend on both force and distance vectors. Calculating these often involves finding the component of force perpendicular to the distance vector (lever arm). Understanding how to decompose vectors into their components is crucial for accurately determining the magnitude and direction of moments and torques in various mechanical systems.

45. Can you explain how vector components are used in navigation and GPS systems?

In navigation and GPS systems, positions and velocities are typically represented as vectors with components in a global coordinate system (like latitude, longitude, and altitude). Understanding how to decompose and transform these vectors is crucial for calculating distances, directions, and relative positions. Vector components also play a key role in dead reckoning and inertial navigation systems.

46. How do vector components help in understanding and calculating fluid dynamics?

In fluid dynamics, vector fields represent quantities like velocity and vorticity. Understanding how these vectors decompose into components is crucial for analyzing flow patterns, calculating pressure gradients, and solving the Navier-Stokes equations. Vector components are also essential in understanding concepts like divergence and curl in fluid flow.

47. What's the significance of vector components in understanding and calculating electromagnetic fields?

In electromagnetism, fields are often represented as vector fields. Understanding how these vectors decompose into components is crucial for analyzing field behavior, calculating potentials, and solving Maxwell's equations. The concept of parallel and perpendicular components is particularly important in understanding wave propagation and polarization of electromagnetic waves.

48. How do vector components help in understanding and implementing neural networks?

In neural networks, weights and activations are often represented as vectors. Understanding how these vectors decompose into components is crucial for implementing backpropagation, analyzing network behavior, and optimizing performance. Vector components also play a key role in techniques like dimensionality reduction and feature extraction in deep learning.