# Vectors: Given z in u+v=z, find u and v (with constraints)

In summary, we are given a vector in R^6 and are asked to find two vectors, u and v, such that their sum is equal to the given vector and u has all equal coordinates while v's coordinates add up to 0. This involves creating a decomposition and deriving a system of equations with 7 variables.

## Homework Statement

Given a vector z=<-12, 1, 1, 2, 7, 0> in R^6
and z=u+v, then find u and v such that u's coordinates are all equal to each other (like <0,0,0,0,0,0>) and v has coordinates that add up to 0

z=u+v

## The Attempt at a Solution

i have no idea how to approach this...is this a problem about direct sums? (since that is what we're studying at the moment) On a related note, what is the difference between regular addition and a direct sum?

Last edited:
u has coordinates equal to 0?

What do you mean with this?

micromass said:
What do you mean with this?

oops sorry i mean the coordinates are all equal

Allright, so you want to make the following decomposition:

$$(-12, 1, 1, 2, 7, 0)=(a,a,a,a,a,a)+(b,c,d,e,f,g)$$

where b+c+d+e+f+g=0.

Now, can you derive a system of equations from this? I claim that you can obtain a system of 7 equations and 7 indeterminates (which are of course a,b,c,d,e,f,g).

## 1. What is a vector?

A vector is a quantity that has both magnitude (size) and direction. It is usually represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

## 2. What is the difference between a vector and a scalar?

A scalar is a quantity that has only magnitude, while a vector has both magnitude and direction. For example, temperature is a scalar quantity while velocity is a vector quantity.

## 3. How do you add or subtract vectors?

To add or subtract vectors, you need to first ensure that they are in the same direction. Then, you can simply add or subtract the magnitudes of the vectors to get the new vector. For example, if u = 2i + 3j and v = -i + 2j, then u+v = (2-1)i + (3+2)j = i + 5j.

## 4. What are the constraints when solving for u and v in the equation z = u+v?

The constraints when solving for u and v in the equation z = u+v are that the vectors u and v must be in the same direction and that their magnitudes must add up to the magnitude of z.

## 5. Can vectors be multiplied?

Vectors can be multiplied, but the result is not a single number. Instead, there are two types of vector multiplication: dot product and cross product. The dot product of two vectors results in a scalar quantity, while the cross product results in a vector quantity.

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