Question

Find the components of the vector with initial point (4, 2) in the direction of the vector [-2, 6] which has a terminal point (0, y) for some y.

a. y = -1
b. y = 1
c. y = 3
d. y = 5

Answer 1

The components of the vector with the initial point (4, 2) in the direction of [-2, 6] are (-4, 12). The y-coordinate of the terminal point is 12 and the possible values for y are -1, 1, 3, and 5.

Step-by-step explanation:

To find the components of the vector in the direction of the vector [-2, 6], we need to find the projection of the vector with initial point (4, 2) onto the given direction vector. The projection of a vector onto another vector can be found using the dot product. The formula for the projection of vector A onto vector B is: ProjBA = (A · B) / |B|.

In this case, we have vector A with initial point (4, 2) and vector B = [-2, 6]. The dot product of A and B is: A · B = (4 * -2) + (2 * 6) = -8 + 12 = 4. The magnitude of B is: |B| = sqrt((-2)^2 + 6^2) = sqrt(4 + 36) = sqrt(40) = 2sqrt(10). Therefore, the projection of A onto B is: ProjBA = (4) / (2sqrt(10)) = (2sqrt(10)) / sqrt(10) = 2.

The components of the vector with initial point (4, 2) in the direction of vector [-2, 6] are: (2 * (-2), 2 * 6) = (-4, 12). Therefore, the vector with terminal point (0, y) has y-coordinate 12 and the possible values for y are a. -1, b. 1, c. 3, d. 5.