Key Concepts and Formulas

• Scalar Projection: The scalar projection of a vector $\overrightarrow{a}$ onto a vector $\overrightarrow{b}$ is given by $\text{Proj}_{\overrightarrow{b}} \overrightarrow{a} = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{b}|}$. This represents the length of the component of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$.
• Vector Dot Product and Triangle Sides: For any three points A, B, and C, the dot product of vectors originating from the same vertex, say $\overrightarrow{AB}$ and $\overrightarrow{AC}$, can be related to the lengths of the sides of triangle ABC using the relation derived from the Law of Cosines: $2(\overrightarrow{AB} \cdot \overrightarrow{AC}) = |\overrightarrow{AB}|^2 + |\overrightarrow{AC}|^2 - |\overrightarrow{BC}|^2$.

Step-by-Step Solution

Step 1: Understand the Goal We are asked to find the projection of the vector $\overrightarrow{AB}$ onto the vector $\overrightarrow{AC}$. This means we need to calculate the scalar projection of $\overrightarrow{AB}$ on $\overrightarrow{AC}$.

Step 2: Identify Given Information We are given the magnitudes of the sides of triangle ABC:

• $|\overrightarrow{BC}| = 8$
• $|\overrightarrow{CA}| = 7$, which implies $|\overrightarrow{AC}| = 7$
• $|\overrightarrow{AB}| = 10$

Step 3: Recall the Formula for Scalar Projection The scalar projection of $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is given by: $\text{Proj}_{\overrightarrow{AC}} \overrightarrow{AB} = \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{|\overrightarrow{AC}|}$

Step 4: Determine the Magnitude of the Projection Vector From the problem statement, the magnitude of $\overrightarrow{AC}$ is directly given: $|\overrightarrow{AC}| = 7$.

Step 5: Calculate the Dot Product $\overrightarrow{AB} \cdot \overrightarrow{AC}$ We can use the relationship derived from the Law of Cosines in vector form. Consider the vector $\overrightarrow{BC} = \overrightarrow{AC} - \overrightarrow{AB}$. Squaring its magnitude: $|\overrightarrow{BC}|^2 = |\overrightarrow{AC} - \overrightarrow{AB}|^2$ $|\overrightarrow{BC}|^2 = (\overrightarrow{AC} - \overrightarrow{AB}) \cdot (\overrightarrow{AC} - \overrightarrow{AB})$ $|\overrightarrow{BC}|^2 = |\overrightarrow{AC}|^2 + |\overrightarrow{AB}|^2 - 2(\overrightarrow{AC} \cdot \overrightarrow{AB})$

Rearranging the formula to solve for the dot product $\overrightarrow{AB} \cdot \overrightarrow{AC}$: $2(\overrightarrow{AB} \cdot \overrightarrow{AC}) = |\overrightarrow{AB}|^2 + |\overrightarrow{AC}|^2 - |\overrightarrow{BC}|^2$ $\overrightarrow{AB} \cdot \overrightarrow{AC} = \frac{|\overrightarrow{AB}|^2 + |\overrightarrow{AC}|^2 - |\overrightarrow{BC}|^2}{2}$

Now, substitute the given magnitudes: $|\overrightarrow{AB}| = 10$, $|\overrightarrow{AC}| = 7$, $|\overrightarrow{BC}| = 8$. $\overrightarrow{AB} \cdot \overrightarrow{AC} = \frac{10^2 + 7^2 - 8^2}{2}$ $\overrightarrow{AB} \cdot \overrightarrow{AC} = \frac{100 + 49 - 64}{2}$ $\overrightarrow{AB} \cdot \overrightarrow{AC} = \frac{149 - 64}{2}$ $\overrightarrow{AB} \cdot \overrightarrow{AC} = \frac{85}{2}$

Step 6: Compute the Scalar Projection Now, substitute the dot product and the magnitude of $\overrightarrow{AC}$ into the projection formula: $\text{Proj}_{\overrightarrow{AC}} \overrightarrow{AB} = \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{|\overrightarrow{AC}|} = \frac{\frac{85}{2}}{7}$ $\text{Proj}_{\overrightarrow{AC}} \overrightarrow{AB} = \frac{85}{2 \times 7}$ $\text{Proj}_{\overrightarrow{AC}} \overrightarrow{AB} = \frac{85}{14}$

Common Mistakes & Tips

• Vector Direction: Be careful with the direction of vectors. While $|\overrightarrow{CA}| = |\overrightarrow{AC}|$, the vectors themselves are opposite. The formula for the dot product uses vectors originating from the same point.
• Projection Formula: Ensure you are calculating the scalar projection ($\frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{b}|}$) and not the vector projection ($\frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|\overrightarrow{b}|^2} \overrightarrow{b}$).
• Law of Cosines Relation: The formula $2(\overrightarrow{u} \cdot \overrightarrow{v}) = |\overrightarrow{u}|^2 + |\overrightarrow{v}|^2 - |\overrightarrow{w}|^2$, where $\overrightarrow{w} = \overrightarrow{v} - \overrightarrow{u}$, is a powerful tool when dealing with side lengths of a triangle and dot products.

Summary To find the projection of vector $\overrightarrow{AB}$ on $\overrightarrow{AC}$, we utilized the scalar projection formula $\frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{|\overrightarrow{AC}|}$. We determined the magnitude $|\overrightarrow{AC}|$ directly from the problem. The dot product $\overrightarrow{AB} \cdot \overrightarrow{AC}$ was calculated using a vector form of the Law of Cosines, relating it to the squares of the magnitudes of the sides of triangle ABC. Substituting these values into the projection formula yielded the final answer.

The final answer is $\boxed{\frac{85}{14}}$ which corresponds to option (C).