Understanding Vector Equations and Commutative Property in Linear Algebra

Vectors play a crucial role in linear algebra, offering a powerful tool for representing physical quantities with both magnitude and direction. In this article, we explore how to solve vector equations and understand the commutative property of vector addition. Let's start with an example.

Solving Vector Equations

Given the vectors u 3i 3j, v 13i - 7j, and 2w 3u - v 4i - 4j, we are to find the coordinates of the vector w.

Substitution and Simplification

First, substitute the values of u and v into the equation:

u 3i 3j

v 13i - 7j

Next, compute 3u and -v:

Add 3u and -v to get:

3u - v (9i 9j) (-13i 7j) -4i 16j

Now substitute this result back into the original equation:

2w (-4i 16j) 4i - 4j

Isolate 2w by subtracting -4i 16j from 4i - 4j:

2w (4i - 4j) - (-4i 16j) 4i - 4j 4i - 16j 8i - 20j

Divide by 2 to solve for w:

w (8i - 20j)/2 4i - 10j

Thus, the coordinates of vector w are:

w (4, -10)

Commutative Property of Vector Addition

The equation u v v u represents the commutative property of vector addition. This property holds true for all vectors in any vector space. It means that the sum of two vectors is invariant with respect to the order in which they are added.

Let's show this with the given vectors:

u 3i 3j, v 13i - 7j

u v (3i 3j) (13i - 7j) 16i - 4j

v u (13i - 7j) (3i 3j) 16i - 4j

As we can see:

u v v u

Hence, the commutative property is always true for the given vectors.