Final answer:
To find the component of vector a = 8i + 9j in the direction of point P(7, 10), one needs to calculate the unit vector in the direction of P and take the dot product of vector a with this unit vector.
Step-by-step explanation:
The task at hand involves finding the component of a vector in a specific direction. Given the vector a = 8i + 9j and the point P(7, 10), we first need to determine the direction vector from the origin to point P, which is P = 7i + 10j. To find the component of a in the direction of P, we must project vector a onto vector P.
The projection of a onto P is given by the dot product of a with the unit vector in the direction of P, divided by the magnitude of P. The unit vector uP is P/|P|, where |P| is the magnitude of P which can be calculated using the Pythagorean theorem as |P| = √(72 + 102). We then find the component of a along uP as (a · uP)uP.
This process involves dot product and basic principles of vector projection, utilizing scalar and vector multiplication to achieve the desired result. It is a valuable skill in high school and college-level physics and engineering.