In Exercises 67-70, find two vectors in opposite directions that are orthogonal to the vector $\mathbf{u}$. (There are many correct answers.) $\mathbf{u}=\frac{1}{2}\mathbf{i}-\frac{2}{3}\mathbf{j}$

Step by Step Solution: Step 1. To find the vectors in opposite directions that are orthogonal to vector u, we need to find the vector v such that u ⋅ v = 0, which means that the dot product of u and v is zero. Step 2. Let the vector v = ai + bj. Then, we need to find a and b such that u ⋅ v = 0. Step 3. Substituting u and v, we get (1/2 * a) + (-2/3 * b) = 0. Step 4. Solving for b, we find that: b = (3/4) * a. Step 5. Therefore, the vector v that is orthogonal to u is v = a * i + (3/4) * a * j = a * (i + (3/4) * j) where a can be any non-zero constant. Step 6. So, two vectors in opposite directions that are orthogonal to u are v1 = a * (i + (3/4) * j) and v2 = -a * (i + (3/4) * j), where a can be any non-zero constant. Step 7. To check that v1 and v2 are orthogonal to u, we can calculate the dot product of u and v1 and the dot product of u and v2 to confirm that both are equal to zero. Final Answer: Two vectors in opposite directions that are orthogonal to u are: v1 = a * (i + (3/4) * j) and v2 = -a * (i + (3/4) * j), where a can be any non-zero constant.