Unit Vector Notation: Solve a1, a2 for Vector Equation

[saber1357]

Homework Statement

Vector a1 + vector a2 = 5*vector a3
Vector a1 - vector a2 = 3*vector a3
Vector a3 = 2i + 2j (i and j are the vector components)
Express 1) vector a1 and 2) vector a2 in unit vector notation

Homework Equations

Vector R = Ax + Yx

The Attempt at a Solution

I took the first equation and replaces the a3 with 2i+2j, so vector a1 + vector a2 = 5(2i + 2j)
a1 + a2 = 10i + 10j
I solved for R and found direction, but I do'nt believe that helps at all. (R = sqr root of 200 and direction was 45 degrees).
From the first equation with addition, I could conclude that vector a1 could equal = 10i +0j and vector a2 could equal 0i + 10j. However, these values don't work for the second equation with the subtraction. Is there some sort of secret to this problem?

 

[learningphysics]

Don't worry about magnitudes and angles for this problem. This one is straight algebra. substitute a3 into the second equation also. That'll give you another equation with a1 and a2.

You have two equations with two unknowns (a1 and a2)... solve those, get a1 and a2.

 

[m2003]

I would approach this problem by first recognizing that the given equations are in vector form, where vector a1 and a2 are added and subtracted, respectively, to form a new vector a3. In order to solve for a1 and a2 in unit vector notation, we can use the properties of vector addition and subtraction.

Starting with the first equation, we can express it in unit vector notation as:

(a1x + a2x)i + (a1y + a2y)j = 5(2i + 2j)

By comparing the coefficients on the left and right sides, we can see that:

a1x + a2x = 10
a1y + a2y = 10

Similarly, for the second equation, we have:

(a1x - a2x)i + (a1y - a2y)j = 3(2i + 2j)

Comparing coefficients again, we can see that:

a1x - a2x = 6
a1y - a2y = 6

Now, we have a system of two equations with two unknowns (a1x, a1y). Solving these equations, we get:

a1x = 8
a1y = 2

Substituting these values back into the original equations, we can solve for a2x and a2y:

a2x = 2
a2y = 8

Therefore, in unit vector notation, we have:

vector a1 = 8i + 2j
vector a2 = 2i + 8j

This satisfies both of the given equations and is the solution for the given vector equation.