Adding Vectors Algebraically & Graphically

Adding vectors algebraically involves combining the corresponding components of the vectors. This operation is essential in both mathematics and applications like physics, engineering, and AI.

1. Vector Representation and Problem Setup

Let’s say you want to add two vectors, a and b. Here’s how we represent these vectors:

Vector a: 𝑎 =[6,−2]

This means that the vector a has a magnitude of 6 along the x-axis and -2 along the y-axis.

Vector b: 𝑏 = [−4,4]

This means that the vector b has a magnitude of -4 along the x-axis and 4 along the y-axis.

2. How to Add Vectors Algebraically

To add vectors a and b, we simply add their corresponding components. For vectors in 2D (as we have here), the addition looks like this:

a+b=[a1 +b1,a2+b2]

Where:

• a1 and a2 are the x and y components of vector a.
• b1 and b2 are the x and y components of vector b.

Now, substitute the values of a and b:

a + b=[6+(−4),−2+4]
a+b=[2,2]

So, the sum of vectors a and b is a+b=[2,2].

3. Geometric Interpretation

In terms of geometry, we can visualize vector addition using the tip-to-tail method:

• Place the tail of vector b at the tip of vector a.
• Draw the resultant vector from the tail of a to the tip of b.

This geometric sum of vectors corresponds to the algebraic sum
[2,2], meaning the new vector moves 2 units in the positive x-direction and 2 units in the positive y-direction.

4. Properties of Vector Addition

- Commutative Property:
The order in which you add vectors doesn't affect the result.

a+b = b+a

- Associative Property:
Vector addition is associative, meaning you can group vectors in any way when adding them.

(a+b)+c=a+(b+c)

- Zero Vector:
Adding the zero vector to any vector doesn’t change the original vector.

a+0=a

- Additive Inverse:
Every vector has an additive inverse that "cancels it out".

a+(−a)=0

6. Conclusion

Adding vectors algebraically is straightforward: you simply add the components of the vectors. Whether you're dealing with 2D, 3D, or higher dimensions, this operation is essential for everything from physics simulations to machine learning. In this case, the sum of vectors a=[6,−2] and b=[−4,4] gives the resultant vector
a+b=[2,2], which you can visualize geometrically and understand algebraically.