Question 21

Explanation

Position Vector of Point A

Let the position vector of point $A$ be $\mathbf{a}$

Since $A$ lies on a circle with center $O$, the vector $\overrightarrow{OA} = \mathbf{a}$ is the radius of the circle.

Tangent Line at Point A

The tangent line at point $A$ is a line that touches the circle at point $A$ and is perpendicular to the radius $OA$ at that point.

Direction Vector of Tangent Line

Let the direction vector of the tangent line at point $A$ be $\mathbf{t}$

Since the tangent line is perpendicular to the radius $OA$ (vector $\mathbf{a}$), the direction vector of the tangent line $\mathbf{t}$ must be perpendicular to vector $\mathbf{a}$

Perpendicular Vector Condition

Two vectors are perpendicular if and only if their dot product is zero

Conclusion

Therefore, to prove that the tangent line at $A$ is perpendicular to the radius $OA$, we need to show that the dot product of vector $\mathbf{a}$ and vector $\mathbf{t}$ is zero

This is the vector representation to prove that the tangent line of the circle at point $A$ is perpendicular to the radius $OA$

The most appropriate answer is C.