2. (a) Given the vectors $$ p=\binom{m+3}{2-n}, \mathbf{q}=\binom{3 m-1}{n-8} $$ and $$ p=q $$, find the values of $$ m $$ and $$ n $$.

Question

UpStudy AI · Accepted Answer

To solve for m and n, we equate the corresponding components of the vectors p and q.

The first component of p is m + 3, and the first component of q is 3m - 1. Setting these equal:
  m + 3 = 3m - 1

Now, solve for m:
  m + 3 = 3m - 1
  3 + 1 = 3m - m       (add m to both sides and add 1)
  4 = 2m
  m = 2

Next, equate the second components. The second component of p is 2 - n, and the second component of q is n - 8:
  2 - n = n - 8

Solve for n:
  2 + 8 = n + n      (add n to both sides and add 8)
  10 = 2n
  n = 5

Thus, the values are:
  m = 2  and n = 5
$$ m = 2 $$ and $$ n = 5 $$.

2. (a) Given the vectors \( p=\binom{m+3}{2-n}, \mathbf{q}=\binom{3 m-1}{n-8} \) and \( p=q \), find the values of \( m \) and \( n \).