To find the coordinates of point $C(5, k)$ such that the vector $\overrightarrow{AC}$ is orthogonal to the vector $\overrightarrow{AB}$, we can use the property that two vectors are orthogonal if and only if their dot product is zero.

Step 1: Find the components of the vectors

We are given:

• Point $A(4, 2)$
• Point $B(-2, 5)$
• Point $C(5, k)$

We need to find the vectors $\overrightarrow{AC}$ and $\overrightarrow{AB}$.

• The vector $\overrightarrow{AC}$ is given by the difference of the coordinates of points $C$ and $A$: $\overrightarrow{AC} = (5 - 4, k - 2) = (1, k - 2)$

• The vector $\overrightarrow{AB}$ is given by the difference of the coordinates of points $B$ and $A$: $\overrightarrow{AB} = (-2 - 4, 5 - 2) = (-6, 3)$

Step 2: Use the condition for orthogonality

For the vectors $\overrightarrow{AC}$ and $\overrightarrow{AB}$ to be orthogonal, their dot product must be zero. The dot product of two vectors $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $\overrightarrow{AC} \cdot \overrightarrow{AB} = x_1 x_2 + y_1 y_2$

Substitute the components of $\overrightarrow{AC}$ and $\overrightarrow{AB}$ into this formula: $(1, k - 2) \cdot (-6, 3) = 1 \times (-6) + (k - 2) \times 3$ $= -6 + 3(k - 2)$ Simplifying: $-6 + 3(k - 2) = -6 + 3k - 6 = 3k - 12$

Step 3: Set the dot product equal to zero

For orthogonality, we set the dot product equal to zero: $3k - 12 = 0$

Step 4: Solve for $k$

Solve the equation: $3k = 12$ $k = 4$

Final Answer:

The coordinates of point $C$ are $(5, 4)$.

Let me know if you would like further details or clarification! Here are some related questions for practice:

1. What does the geometric interpretation of orthogonality mean for two vectors?
2. How would you calculate the distance between points $A$ and $C$?
3. How can you prove that two vectors are parallel?
4. How do you find the angle between two vectors?
5. How does the concept of dot product relate to work in physics?

Tip: When working with vectors, always remember that the dot product is zero when the vectors are orthogonal (perpendicular), which is key to many geometric problems!