Vectors are used in a range of mathematical and scientific applications, and they play an important role in physics, engineering, and computer science. A vector is a quantity that has both magnitude and direction, and it can be represented graphically as an arrow. In this article, we will explore how to express a vector as a linear combination of other vectors.
What Is A Linear Combination?
A linear combination is a sum of scalar multiples of vectors. In other words, if we have two vectors, A and B, and two scalars, c and d, then the linear combination of A and B can be expressed as:
cA + dB
where c and d are scalars. This means that we can take any two vectors, multiply each one by a scalar, and add them together to create a new vector.
Expressing A Vector As A Linear Combination
Now that we understand what a linear combination is, let’s look at how to express a vector as a linear combination of other vectors. Suppose we have three vectors, A, B, and C, and we want to express vector C as a linear combination of vectors A and B. In other words, we want to find scalars x and y such that:
C = xA + yB
To find the values of x and y, we need to solve a system of equations. We can do this by using the dot product of vectors. The dot product of two vectors A and B is defined as:
A · B = |A||B|cosθ
where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them. We can use the dot product to find the value of x and y in our equation:
C · A = xA · A + yB · A
C · B = xA · B + yB · B
We can rewrite these equations as:
x = (C · A – yB · A)/|A|^2
y = (C · B – xA · B)/|B|^2
where |A|^2 and |B|^2 are the magnitudes of vectors A and B squared.
Example
Let’s look at an example to see how this works. Suppose we have three vectors:
A = (2, 3)
B = (1, -1)
C = (5, 2)
We want to express vector C as a linear combination of vectors A and B. First, we need to find the dot products:
C · A = (5)(2) + (2)(3) = 16
C · B = (5)(1) + (2)(-1) = 3
Next, we can use the equations we derived earlier to find the values of x and y:
x = (C · A – yB · A)/|A|^2
y = (C · B – xA · B)/|B|^2
Substituting in the dot products and magnitudes:
x = (16 – 3y)/13
y = (3 – 2x)/2
We can solve this system of equations to find that:
x = 1
y = 3
Therefore, we can express vector C as a linear combination of vectors A and B:
C = 1A + 3B
Applications
Expressing a vector as a linear combination of other vectors is a useful technique that has many applications. For example, it can be used to solve problems in physics, such as finding the equilibrium position of a system of forces. It can also be used in computer graphics to transform objects in 3D space.
Conclusion
In this article, we have explored how to express a vector as a linear combination of other vectors. We have seen that this involves solving a system of equations using the dot product of vectors. This technique has many applications in physics, engineering, and computer science, and it is an important tool for solving problems in these fields.