For what values of a and b are these two vectors collinear (linear algebra)?

[BlueRope]

Homework Statement

For what values of a and b are these two vectors collinear (linear algebra)?

t = (3a -4b +5, 5b -2a -8)
v = (-3, 4)

t and v are vectors

Homework Equations

The Attempt at a Solution

I'd like to know what the steps are. I don't really care about the answer.

 

[alanlu]

t and v are collinear if t = kv for some scalar k. Expand and solve for a and b in terms of k.

 

[BlueRope]

I was thinking of doing that, until I saw the answer, which was t = 7(a-1)(-3,4) or 7(a-1)vectorz.

 

[BlueRope]

Basically, he wrote 6a -b -4 = 0

then he isolated b, which gives:

b = 6a -4

then he somehow came with

the answer, which is, like I said:

t = 7(a-1)(-3,4) or 7(a-1)vector z.

I have no idea what the teacher did.

 

[alanlu]

Another way of determining vectors are collinear is to see if the dot product of one of the vectors and basis vectors of the other vector's orthogonal space evaluate to 0. In the plane, a vector v's orthogonal space has one dimension and is in fact perp v.

It appears that the equation t * perp v = 0 was solved for b here.

 

[BlueRope]

I finally understood how he did it. However, I don't understand how it answers the question. He wanted the values of b and a, but we only got the value of k.

If the answer is 7(a-1) vector z then a GENUINE answer would be: b = 0 and a = R or any value, right?

 

[alanlu]

I don't understand where 7(a-1) vector z is from here, but letting a = R gives b = 6R - 4. Values a, b such that b = 6a -4 should be enough to answer this question.