Question

let a=⟨1,4⟩ and b=⟨4,−3⟩ . show that there are scalars s and t so that sa tb=⟨0,−19⟩ you might want to sketch the vectors to get some intuition. s= t=

Answer 1

The scalars s = -4 and t = 1 when multiplied by vectors a = ⟨1,4⟩ and b = ⟨4,−3⟩ respectively, will yield the vector ⟨0,−19⟩ as required by the problem statement in vector algebra.

Step-by-step explanation:

The student is dealing with a problem in vector algebra where they need to find scalars s and t such that when vector a is scaled by s and vector b is scaled by t, the resultant vector is ⟨0,−19⟩. We use the given vectors a=⟨1,4⟩ and b=⟨4,−3⟩ in this context.

To find the correct scalars, we set up the equation sa + tb = ⟨0,−19⟩ and solve for s and t. This translates to the system of equations:

• 1s + 4t = 0
• 4s - 3t = -19

To find the values of s and t, we can use methods such as substitution or elimination. For this system, the elimination method might be more straightforward. Multiplying the first equation by 4 gives:

• 4s + 16t = 0

Subtracting this from the second equation gives:

• -19t = -19

Therefore, t = 1. Substituting t into the first equation gives:

• s + 4(1) = 0
• s = -4

Then we have found that s = -4 and t = 1.