Gandalf the grey started in the forest of mirkwood at a point p with coordinates (3, 0) and arrived in the iron hills at the point q with coordinates (5, 5). if he began walking in the direction of the vector v - 3i + 2j and changes direction only once, when he turns at a right angle, what are the coordinates of the point where he makes the turn?

Turning point has coordinates

Step-by-step explanation:

Gandalf the Grey started in the Forest of Mirkwood at a point P(3, 0) and began walking in the direction of the vector The coordinates of the vector v are (-3,2). Then he changed the direction at a right angle, so he was walking in the direction of the vector (vectors u and v are perpendicular).

Let B(x,y) be the turning point. Find vectors PB and BQ:

Note that vectors v and PB and vectors u and BQ are collinear, so

Hence

Now solve the system of two equations:

Multiply the first equation by 3, the second equation by 2 and add them:

Substitute it into the first equation:

Turning point has coordinates

the expression (1.26)2t reveals the approximate rate of increases in the number of plants semiannually.

hope this : )

18.8cm hope i at all

step-by-step explanation: