Linear Combination of Vectors

Edited By Komal Miglani | Updated on Jul 02, 2025 07:44 PM IST

Before studying about linear combination of vectors, let's revise about vectors. 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. A vector is represented by a directed line segment (an arrow). In real life, we use vectors for tracking objects that are in motion, and localization of places and things.

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Linear Combination of Vectors
Test of collinearity of three points
Important point
Solved Examples Based on Linear Combination of Vectors

Linear Combination of Vectors

In this article, we will cover the concept of Linear Combination of Vectors. 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 five questions have been asked on this topic in JEE Main from 2013 to 2023 including one in 2022.

Linear Combination of Vectors

A vector $\vec{r}$ is said to be a linear combination of vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3 \ldots \ldots, \vec{a}_n$ if there exist scalars $\lambda_1, \lambda_2, \lambda_3, \ldots \ldots, \lambda_n$ such that

$
\vec{r}=\lambda_1 \vec{a}_1+\lambda_2 \vec{a}_2+\lambda_3 \vec{a}_3+\ldots \ldots+\lambda_n \vec{a}_n
$
For example:
vectors $\vec{r}_1=\vec{a}+2 \vec{b}+3 \vec{c}$ and $\vec{r}_2=\vec{a}-4 \vec{b}-9 \vec{c}$ are linear combination of the vectors $\vec{a}, \vec{b}$ and $\vec{c}$.

Linear Independent Vectors
A system of vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3 \ldots \ldots, \vec{a}_n$ is said to be linearly independent, if

$
\begin{array}{ll}
& \lambda_1 \vec{a}_1+\lambda_2 \vec{a}_2+\lambda_3 \vec{a}_3+\ldots \ldots+\lambda_n \vec{a}_n=0 \\
\Rightarrow \quad & \lambda_1=\lambda_2=\lambda_3=\ldots \ldots=\lambda_n=0
\end{array}
$

It can be easily verified that

1. A pair of non-collinear vectors ( say a₁ and a₂ ) are linearly independent.

Let $\vec{a}_1$ and $\vec{a}_2$ are two non-collinear vectors such that $\lambda_1 \vec{a}_1+\lambda_2 \vec{a}_2=0$.
Let, $\quad \lambda_1, \lambda_2 \neq 0$
$\Rightarrow \quad \vec{a}_1=-\frac{\lambda_2}{\lambda_1} \vec{a}_2$
Now, $-\frac{\lambda_2}{\lambda_1}$ is a scalar, because $\lambda_1$ and $\lambda_2$ are scalars.
Hence, eq (i) expresses $\vec{a}_1$ as product of $\vec{a}_2$ by a scalar, so that $\vec{a}_1$ and $\vec{a}_2$ are collinear.
Which contradict the given fact because $\overrightarrow{a_1}$ and $\overrightarrow{a_2}$ are given to be non - collinear.
Th thus, our supposition that $\lambda_1 \neq 0$ and $\lambda_2 \neq 0$ is wrong.
Hence, $\lambda_1=0$ and $\lambda_2=0$

2. A triad of non-coplanar vector is linearly independent

If a, b, c are three non-zero, non-coplanar vectors and x, y, z are three scalars such that

$\begin{array}{r}x a+y b+z c=0 \\ \text { Then, } x=y=z=0\end{array}$

Proof: It is given that $\mathrm{xa}+\mathrm{yb}+\mathrm{zc}=0$
Suppose that $x \neq 0$
Then Eq. (i) can be written as

$
\begin{gathered}
x=-y b-z \mathrm{c} \\
\Rightarrow \mathrm{a}=-\frac{y}{x} \mathrm{~b}-\frac{z}{x} \mathrm{c} \quad \ldots \text { (ii) }
\end{gathered}
$
Now, $\frac{y}{x}$ and $\frac{z}{x}$ are scalars because $x, y$ are scalars. Thus, Eq. (ii) expresses an as a linear combination of $b$ and $c$. Hence, a is coplanar with b and c which is contrary to our hypothesis because a, b, and c are given to be non-coplanar. Thus, our supposition that $x \neq 0$ is wrong.

Hence, $x=0$.
Similarly, we can prove that $\mathrm{y}=0$ and $\mathrm{z}=0$.

Note: 4 vectors are always linearly dependent

A system of vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3 \ldots \ldots, \vec{a}_n$ is said to be linearly dependent, if there exists scalars $\lambda_1, \lambda_2, \lambda_3, \ldots \ldots \ldots, \lambda_{\mathrm{n}}$ not all zero such that

$
\lambda_1 \vec{a}_1+\lambda_2 \vec{a}_2+\lambda_3 \vec{a}_3+\ldots \ldots+\lambda_n \vec{a}_n=0
$
It can be easily verified that

1. A pair of collinear vectors is linearly dependent.

2. A triad of coplanar vectors is linearly dependent.

Test of collinearity of three points

Let $A(\vec{a}), B(\vec{b})$ and $C(\vec{c})$, be three points in space
If we can find $x, y, z$, not all zero such that
(i) $x \vec{a}+y \vec{b}+z \vec{c}=0$ and (ii) $x+y+z=0$

Then the points $A, B$, and $C$ are collinear

Proof:

Let us suppose that points $A, B$ and $C$ are collinear and their position vectors are :, $\vec{b}$ and $\vec{c}$ respectively. Let $C$ divide the join of $\vec{a}$ and $\vec{b}$ in the ratio $y: x$
Then,

$
\begin{aligned}
& \vec{c}=\frac{x \vec{a}+y \vec{b}}{x+y} \\
& x \vec{a}+y \vec{b}-(x+y) \vec{c}=\overrightarrow{0} \\
& x \vec{a}+y \vec{b}+z \vec{c}=\overrightarrow{0}, \text { where } z=-(x+y) \\
& x+y+z=x+y-(x+y)=0
\end{aligned}
$
Hence, the three points $A, B$, and $C$ are collinear.

Theorem 1:

If $\vec{a}$ and $\vec{b}$ are two non-zero, non-collinear vectors, then every vector $\vec{r}$ coplanar with $\vec{a}$ and $\vec{b}$ can be expressed in one and only one way as a linear combination $x \vec{a}+y \vec{b}$ where $x$ and $y$ being scalars.

Proof:

Let $O$ be any point such that $\overrightarrow{O A}=\vec{a}$ and $\overrightarrow{O B}=\vec{b}$ As $\vec{r}$ is coplanar with $\vec{a}$ and $\vec{b}$, the lines $O A, O B$ and $O R$ are coplanar. Through $R$, draw lines parallel to $O A$ and $O B$, meeting them at $P$ and $Q$, respectively. Clearly,

$
\overrightarrow{\longrightarrow P}=x \overrightarrow{O A}=x \vec{a} \quad(\because \overrightarrow{O P} \text { and } \overrightarrow{\longrightarrow A} \text { are collinear vectors })
$
Also,

$
\overrightarrow{O Q}=y \overrightarrow{O B}=y \vec{b} \quad(\because \overrightarrow{O Q} \text { and } \overrightarrow{O B} \text { are collinear vectors })
$
$
\begin{aligned}
\vec{r} & =\overrightarrow{O R}=\overrightarrow{O P}+\overrightarrow{P R}=\overrightarrow{O P}+\overrightarrow{O Q} \quad(\because \overrightarrow{O Q} \text { and } \overrightarrow{P R} \text { are equal }) \\
& =x \vec{a}+y \vec{b}
\end{aligned}
$
Thus, $\vec{r}$ can be expressed in one way as a linear combination $x \vec{a}+y \vec{b}$.
Theorem 2
If $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar vectors, then any vector $\vec{r}$ can be uniquely expressed as a linear combination $\vec{x} \vec{a}+y \vec{b}+z \vec{c}$ where $x, y$ and $z$ being scalars.

NOTE:
1.

If vectors $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$ and $\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ are coplanar, then

$
\left|\begin{array}{lll}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|=0
$

(Proof of this will be seen in the concept of Scalar triple Product)
2.

If vectors $x_1 \vec{a}+y_1 \vec{b}+z_1 \vec{c}, \quad x_2 \vec{a}+y_2 \vec{b}+z_2 \vec{c}$ and $x_3 \vec{a}+y_3 \vec{b}+z_3 \vec{c}$ are coplanar where $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar. Then,

$
\left|\begin{array}{lll}
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
x_3 & y_3 & z_3
\end{array}\right|=0
$

Important point

1) Two collinear vectors are always linearly dependent.

2) Two non-collinear non-zero vectors are always linearly independent.

3) Three coplanar vectors are always linearly dependent.

4) Three non-coplanar non-zero vectors are always linearly independent.

5) More than three vectors are always linearly dependent.

Solved Examples Based on Linear Combination of Vectors

Example 1: Let the vectors $\overrightarrow{\mathrm{a}}=(1+\mathrm{t}) \hat{\mathrm{i}}+(1-\mathrm{t}) \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=(1-\mathrm{t}) \hat{\mathrm{i}}+(1+\mathrm{t}) \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\mathrm{t} \hat{\mathrm{i}}-\mathrm{t} \hat{\mathrm{j}}+\hat{\mathrm{k}}, \mathrm{t} \in \mathbf{R}$ be such that for $\alpha, \beta, \gamma \in \mathbf{R}, \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\overrightarrow{0} \Rightarrow \alpha=\beta=\gamma=0$. Then, the set of all values of t is: [JEE MAINS 2022]

Solution

By its given condition $\vec{a}, \vec{b}, \vec{c}$ are linearly independent vectors

$
\begin{aligned}
& \Rightarrow[\vec{a} \vec{b} \vec{c}] \neq 0 \cdots(1) \\
& \text { Now }[\vec{a} \vec{b} \vec{c}] \\
& =\left|\begin{array}{ccc}
1+t & 1-t & 1 \\
1-t & 1+t & 2 \\
t & -t & 1
\end{array}\right|
\end{aligned}
$
$
\begin{aligned}
& c_2 \rightarrow c_1+c_2 \\
& \left|\begin{array}{ccc}
1+t & 2 & 1 \\
1-t & 2 & 2 \\
t & 0 & 1
\end{array}\right|=2\left|\begin{array}{ccc}
1+t & 1 & 1 \\
1-t & 1 & 2 \\
t & 0 & 1
\end{array}\right| \\
& =2[(1+t)-(1-t)+t] \\
& =2[3 \mathrm{t}]=6 \mathrm{t} \\
& {[\tilde{\mathrm{a}} \tilde{\mathrm{b}} \tilde{\mathrm{c}}] \neq 0 \Rightarrow \mathrm{t} \neq 0}
\end{aligned}
$
Hence, the answer is equal to $\mathbf{R}-\{0\}$

Example 2: If $\vec{a}=2 \vec{p}+3 \vec{q}-\vec{r}, \vec{b}=\vec{p}-2 \vec{q}+2 \vec{r}, \vec{c}=-2 \vec{p}+\vec{q}-2 \vec{r}$ and $\vec{R}=3 \vec{p}-\vec{q}+2 \vec{r}$, then $\vec{R}$ equals ( $\vec{p}, \vec{q}, \vec{r}$ are non-coplanar)
Solution: Linear combination of vectors- $\vec{r}=l \vec{a}+m \vec{b}+n \vec{c}$
Any vectors $\vec{r}$ in space can be written as a linear combination of 3 non-coplanar vectors.

$
\begin{aligned}
& \text { Let } \vec{R}=x \vec{a}+y \vec{b}+z \vec{c} \\
& \Rightarrow 3 \vec{p}-\vec{q}+2 \vec{r}=(2 x+y-2 z) \vec{p}+(3 x-2 y+z) \vec{q}+(-x+2 y-2 z) \vec{r}
\end{aligned}
$

$\because \vec{p}, \vec{q}$, and $\gamma$ are non-coplanar, so
$\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}=\overrightarrow{0}$ only when $\alpha=0, \beta=0, \gamma=0$

$
2 x+y-2 z=3 ; 3 x-2 y+z=-1 ;-x+2 y-2 z=2
$

on solving, we get

$
x=2, y=5, z=3
$
$
\therefore \vec{R}=2 \vec{a}+5 \vec{b}+3 \vec{c}
$
Hence, the answer is $2 \vec{a}+5 \vec{b}+3 \vec{c}$

Example 3: $3 \lambda c+2 \mu(a \times b)=0$ then
Solution: $3 \lambda \vec{c}+2 \mu(\vec{a} \times \vec{b})=0$
Where $\lambda$ and $\mu$ are scaler constant
$3 \lambda \vec{c}=-2 \mu(\vec{a} \times \vec{b})$
On comparing vectors
We can say that $\vec{c}=-(\vec{a} \times \vec{b})$
Hence
$3 \lambda \vec{c}=-2 \mu(\vec{c})$
Or
$3 \lambda=-2 \mu$
$3 \lambda+2 \mu=0$
Hence, the answer is $3 \lambda+2 \mu=0$

Example 4: Let $\vec{r}=\hat{i}-\hat{k}, \vec{a}=\hat{i}-\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-3 \hat{j}+4 \hat{k}$, then linear combinations of $\vec{a}$ and $\vec{b}$ equal $\vec{r}$ ?
Solution

$
3 \vec{a}-\vec{b}=3(\hat{i}-\hat{j}+\hat{k})-(2 \hat{i}-3 \hat{j}+4 \hat{k})=\hat{i}-\hat{k}=\vec{r}
$
Hence, the answer is $\vec{r}=3 \vec{a}-\vec{b}$

Frequently Asked Questions (FAQs)

1. Can linear combinations be used to test for linear independence?

Yes, linear combinations are crucial for testing linear independence. If you can express one vector as a linear combination of others (excluding the trivial combination where all scalars are zero), then the set of vectors is linearly dependent.

2. What's the relationship between linear combinations and vector spaces?

A vector space is closed under linear combinations, meaning any linear combination of vectors in the space will result in another vector in the same space. This is a fundamental property of vector spaces.

3. How do linear combinations relate to basis vectors?

Basis vectors are a set of linearly independent vectors that can be used to express any vector in the space as a unique linear combination. They form the "building blocks" of the vector space.

4. What's the geometric interpretation of a linear combination?

Geometrically, a linear combination represents a point in space that can be reached by following the directions and magnitudes of the component vectors, scaled by their respective scalars.

5. Can you have a linear combination of more than two vectors?

Absolutely! A linear combination can involve any number of vectors. It's simply the sum of each vector multiplied by its corresponding scalar: c₁v₁ + c₂v₂ + c₃v₃ + ... + cₙvₙ.

6. What is a linear combination of vectors?

A linear combination of vectors is the sum of two or more vectors, each multiplied by a scalar (number). It's like combining different ingredients in a recipe, where each vector is an ingredient and the scalar is the amount of that ingredient.

7. How do you calculate a linear combination of vectors?

To calculate a linear combination of vectors, multiply each vector by its corresponding scalar, then add all the resulting vectors together. For example, if you have vectors a and b, and scalars c and d, the linear combination would be ca + db.

8. Can a linear combination result in a zero vector?

Yes, a linear combination can result in a zero vector. This happens when the vectors in the combination cancel each other out completely. For instance, 2a - 2a = 0, where a is any vector.

9. What's the difference between linear combination and vector addition?

Vector addition is a special case of linear combination where all scalars are 1. In linear combination, you can multiply vectors by any scalar before adding them, giving you more flexibility in combining vectors.

10. How does linear combination relate to the concept of span?

The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it's all the vectors you can create by combining the original vectors in different ways.

11. Can a linear combination have negative scalars?

Yes, scalars in a linear combination can be positive, negative, or zero. Negative scalars effectively reverse the direction of the vector they're multiplying.

12. How do you determine if a vector is a linear combination of others?

To determine if a vector v is a linear combination of vectors a, b, and c, you need to solve the equation v = xa + yb + zc for scalars x, y, and z. If a solution exists, then v is a linear combination of a, b, and c.

13. Can linear combinations be used in higher dimensions?

Yes, linear combinations work in any number of dimensions. They're just as applicable to 2D vectors as they are to vectors in 3D, 4D, or even infinite-dimensional spaces.

14. How do linear combinations relate to vector decomposition?

Vector decomposition is the inverse process of linear combination. It involves expressing a given vector as a linear combination of other vectors, typically basis vectors.

15. How does linear combination apply to solving systems of linear equations?

In solving systems of linear equations, each equation can be thought of as a vector. Finding a solution is equivalent to finding a linear combination of these vectors that equals the zero vector.

16. What's the difference between a linear combination and a linear transformation?

A linear combination combines multiple vectors into a single vector, while a linear transformation is a function that maps one vector to another while preserving vector addition and scalar multiplication.

17. What's the connection between linear combinations and matrix multiplication?

Matrix multiplication can be viewed as a series of linear combinations. Each column of the resulting matrix is a linear combination of the columns of the first matrix, with coefficients from the corresponding column of the second matrix.

18. How do linear combinations relate to eigenvalues and eigenvectors?

Eigenvectors are special vectors that, when transformed by a linear transformation, result in a scalar multiple of themselves. This scalar is the eigenvalue, and it represents how the linear transformation scales the eigenvector.

19. Can you have a linear combination of matrices?

Yes, just like vectors, matrices can be combined linearly. A linear combination of matrices is the sum of each matrix multiplied by a scalar.

20. How do linear combinations apply to polynomial spaces?

In polynomial spaces, linear combinations allow us to create new polynomials from existing ones. For example, any polynomial can be expressed as a linear combination of monomials (x^n).

21. What's the significance of linear combinations in signal processing?

In signal processing, complex signals are often decomposed into linear combinations of simpler signals. This is the basis for techniques like Fourier analysis and wavelet transforms.

22. How do linear combinations apply to probability theory?

In probability theory, linear combinations are used to combine random variables. The expected value of a linear combination of random variables is the linear combination of their expected values.

23. What's the role of linear combinations in cryptography?

Linear combinations are used in various cryptographic algorithms, particularly in linear cryptanalysis and in the design of certain types of ciphers like linear feedback shift registers.

24. How do linear combinations apply to color theory?

In color theory, any color can be represented as a linear combination of primary colors. This is the basis for color mixing in both additive (light) and subtractive (pigment) color systems.

25. How do linear combinations relate to the concept of linear operators?

Linear operators preserve linear combinations. If T is a linear operator, then T(av + bw) = aT(v) + bT(w) for any vectors v and w and scalars a and b.

26. Can linear combinations be used to describe rotations in 3D space?

While rotations themselves are not linear combinations, they can be described using linear combinations of basis rotations or through the use of rotation matrices.

27. How do linear combinations relate to the concept of linear codes in coding theory?

Linear codes are sets of codewords that are closed under linear combinations. This property allows for efficient encoding and decoding algorithms.

28. Can linear combinations be used to describe chemical bonding?

Yes, in molecular orbital theory, molecular orbitals are often described as linear combinations of atomic orbitals (LCAO method).

29. What's the connection between linear combinations and Markov chains?

In Markov chains, the probability distribution of future states can be expressed as a linear combination of the current state probabilities, weighted by transition probabilities.

30. How do linear combinations apply to sound synthesis?

In sound synthesis, complex waveforms can be created by taking linear combinations of simpler waveforms. This is the basis of additive synthesis in music production.

31. How do linear combinations apply to the study of differential equations?

In the study of differential equations, the principle of superposition states that for linear differential equations, any linear combination of solutions is also a solution. This is a powerful tool for solving complex differential equations.

32. What's the importance of linear combinations in physics?

In physics, linear combinations are used to describe superposition of waves, quantum states, and forces. They allow physicists to combine simpler states or forces to describe more complex phenomena.

33. How do linear combinations apply to computer graphics?

In computer graphics, linear combinations are used for various purposes, including blending colors, interpolating between keyframes in animation, and transforming 3D objects.

34. Can you have a linear combination of functions?

Yes, the concept of linear combination extends to functions as well. For example, Fourier series express periodic functions as linear combinations of sine and cosine functions.

35. What's the role of linear combinations in machine learning?

In machine learning, linear combinations are fundamental to many algorithms. For instance, in linear regression, the predicted output is a linear combination of input features.

36. What's the connection between linear combinations and linear dependence?

A set of vectors is linearly dependent if one of the vectors can be expressed as a linear combination of the others. If no such combination exists (except the trivial case where all scalars are zero), the vectors are linearly independent.

37. What's the relationship between linear combinations and vector projections?

Vector projection can be viewed as finding the linear combination of a unit vector that best approximates another vector. The scalar in this combination is the dot product of the two vectors.

38. How do linear combinations relate to the concept of subspaces?

A subspace is a subset of a vector space that is closed under linear combinations. This means any linear combination of vectors in the subspace will result in another vector in the same subspace.

39. Can linear combinations be used to describe planes and lines in 3D space?

Yes, planes and lines in 3D space can be described using linear combinations. A line is all linear combinations of a single vector (plus a point), while a plane is all linear combinations of two non-parallel vectors (plus a point).

40. How do linear combinations relate to the concept of linear independence in quantum mechanics?

In quantum mechanics, linear combinations of quantum states represent superposition. Linear independence of these states is crucial for understanding the possible outcomes of measurements.

41. Can linear combinations be used to describe chemical reactions?

Yes, chemical equations can be viewed as linear combinations of chemical species. Balancing chemical equations is essentially finding the right scalars for these linear combinations.

42. What's the connection between linear combinations and coordinate systems?

Coordinate systems provide a way to express any vector as a unique linear combination of basis vectors. The coordinates of a vector are the scalars in this linear combination.

43. What's the role of linear combinations in optimization problems?

Many optimization problems involve finding the best linear combination of certain variables to maximize or minimize an objective function, subject to constraints.

44. What's the significance of linear combinations in data compression?

Many data compression techniques, such as principal component analysis (PCA), involve representing data points as linear combinations of a smaller set of basis vectors.

45. How do linear combinations apply to electrical circuits?

In electrical circuit analysis, linear combinations are used to describe current and voltage relationships, particularly in applying Kirchhoff's laws and in superposition of circuit elements.

46. What's the role of linear combinations in computer vision?

In computer vision, linear combinations are used in various tasks such as image blending, feature extraction, and in certain types of neural network layers.

47. What's the role of linear combinations in financial portfolio theory?

In portfolio theory, an investment portfolio can be viewed as a linear combination of different assets, where the scalars represent the proportion of investment in each asset.

48. How do linear combinations relate to the concept of basis change in linear algebra?

Changing the basis of a vector space involves expressing vectors as linear combinations of a new set of basis vectors. This process is fundamental in many applications of linear algebra.

49. Can linear combinations be used to describe interference patterns in physics?

Yes, interference patterns in waves (like light or sound) can be described as linear combinations of the individual waves, where the amplitudes add or subtract based on phase relationships.

50. What's the significance of linear combinations in control theory?

In control theory, linear combinations are used to describe the state of a system and how it evolves over time. They're also used in designing control laws for linear systems.