Question

Find a vector of magnitude 11 in the direction opposite to that of →PQ, where P and Q are the points (1,3,2) and (1,0,8) respectively.

Answer 1

To find a vector of magnitude 11 in the opposite direction of vector PQ, we first calculate the vector PQ and then find the unit vector in the opposite direction. Finally, we multiply the unit vector by the desired magnitude to obtain the vector in the opposite direction.

Step-by-step explanation:

In order to find a vector of magnitude 11 in the opposite direction of →PQ, we first need to calculate the vector →PQ. →PQ can be found by subtracting the coordinates of point P from the coordinates of point Q: →PQ = (1-1, 0-3, 8-2) = (0, -3, 6).

Next, we need to find the unit vector in the opposite direction of →PQ. The magnitude of →PQ is ∣→PQ∣ = sqrt(0^2 + (-3)^2 + 6^2) = ∣→PQ∣ = 7.74. The unit vector is then obtained by dividing →PQ by its magnitude:

→u = →PQ / ∣→PQ∣ = (0/7.74, -3/7.74, 6/7.74) = (0, -0.39, 0.78).

Finally, to find the vector of magnitude 11 in the opposite direction of →PQ, we multiply the unit vector →u by the desired magnitude:

→v = 11 →u = 11(0, -0.39, 0.78) = (0, -4.29, 8.58).