Final answer:
To find the vector component of u that is parallel to v, we can use the dot product formula and trigonometric functions. The dot product gives us the angle between the vectors, and using that angle, we can find the parallel component of u.
Step-by-step explanation:
To find the vector component of u that is parallel to v, we can use the dot product formula. The dot product of two vectors A and B is given by A · B = |A| |B| cosθ, where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. In this case, we want to find the component of u that is parallel to v, so we can write the equation as u_parallel = |u| cosθ.
To find θ, we can use the fact that the dot product of two parallel vectors is equal to the product of their magnitudes: u · v = |u| |v| cosθ. Rearranging the equation, we get cosθ = (u · v) / (|u| |v|). Substituting the given values, we can find cosθ. Then, we can use the formula u_parallel = |u| cosθ to find the parallel component of u.
For example, if u = (3, 4) and v = (1, 2), we have |u| = √(3^2 + 4^2) = 5 and |v| = √(1^2 + 2^2) = √5. The dot product of u and v is u · v = 3(1) + 4(2) = 11. Substituting these values into the equation, we get cosθ = 11 / (5√5). Then, we can find the parallel component of u using u_parallel = 5 cosθ.