Determine whether $\mathbf{v}$ and $\mathbf{w}$ are orthogonal. \[ \mathbf{v}=7 \mathbf{i}+7 \sqrt{3} \mathbf{j}, \mathbf{w}=2 \sqrt{3} \mathbf{i}-\mathbf{j} \] Choose the correct answer below. Not orthogonal Orthogonal

Step 1 ：We are given two vectors, \(\mathbf{v}=7 \mathbf{i}+7 \sqrt{3} \mathbf{j}\) and \(\mathbf{w}=2 \sqrt{3} \mathbf{i}-\mathbf{j}\). We need to determine whether these vectors are orthogonal.

Step 2 ：Two vectors are orthogonal if their dot product is zero. The dot product of two vectors \(\mathbf{v}=a \mathbf{i}+b \mathbf{j}\) and \(\mathbf{w}=c \mathbf{i}+d \mathbf{j}\) is given by \(ac+bd\).

Step 3 ：So, we need to calculate the dot product of \(\mathbf{v}\) and \(\mathbf{w}\) and check if it is zero.

Step 4 ：\(\mathbf{v}\) = [7, 7\(\sqrt{3}\)] and \(\mathbf{w}\) = [2\(\sqrt{3}\), -1]

Step 5 ：The dot product of \(\mathbf{v}\) and \(\mathbf{w}\) is calculated as \((7 * 2\sqrt{3}) + (7\sqrt{3} * -1)\), which is not equal to zero.

Step 6 ：Since the dot product of vectors \(\mathbf{v}\) and \(\mathbf{w}\) is not zero, they are not orthogonal.

Step 7 ：Final Answer: The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are \(\boxed{\text{Not orthogonal}}\).