To determine which of the given vectors are orthogonal to [tex]\((2,1)\)[/tex], we need to check the dot product of each vector with [tex]\((2,1)\)[/tex]. Two vectors are orthogonal if their dot product is zero.
The dot product of two vectors [tex]\((a, b)\)[/tex] and [tex]\((c, d)\)[/tex] is given by:
[tex]\[ a \cdot c + b \cdot d \][/tex]
Let's calculate the dot product for each vector with [tex]\((2,1)\)[/tex]:
1. For vector [tex]\((-3,6)\)[/tex]:
[tex]\[
2 \cdot (-3) + 1 \cdot 6 = -6 + 6 = 0
\][/tex]
The dot product is 0, so [tex]\((-3,6)\)[/tex] is orthogonal to [tex]\((2,1)\)[/tex].
2. For vector [tex]\((1,2)\)[/tex]:
[tex]\[
2 \cdot 1 + 1 \cdot 2 = 2 + 2 = 4
\][/tex]
The dot product is 4, so [tex]\((1,2)\)[/tex] is not orthogonal to [tex]\((2,1)\)[/tex].
3. For vector [tex]\((1,-2)\)[/tex]:
[tex]\[
2 \cdot 1 + 1 \cdot (-2) = 2 - 2 = 0
\][/tex]
The dot product is 0, so [tex]\((1,-2)\)[/tex] is orthogonal to [tex]\((2,1)\)[/tex].
4. For vector [tex]\((-2,-3)\)[/tex]:
[tex]\[
2 \cdot (-2) + 1 \cdot (-3) = -4 - 3 = -7
\][/tex]
The dot product is -7, so [tex]\((-2,-3)\)[/tex] is not orthogonal to [tex]\((2,1)\)[/tex].
Thus, the vectors that are orthogonal to [tex]\((2,1)\)[/tex] are:
- Option A: [tex]\((-3,6)\)[/tex]
- Option C: [tex]\((1,-2)\)[/tex]
