Gandalf the grey started in the forest of mirkwood at a point with coordinates (2,1) and arrived in the iron hills at the point with coordinates (3,5). if he began walking in the direction of the vector v=4i+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.

(22/5, 11/5)

Step-by-step explanation:

Gandalf starts walking in the vector v = 4i + 2j, the slope of it must be 2/4 = 1/2.

So, calling the turn point (x,y) it must be in that vector, and the start point (2,1) must also be in this line, so:

slope = Δy/Δx

1/2 = (y - 1)/(x - 2)

2y - 2 = x - 2

2y = x

y = x/2

The line that makes a right angle with another is called perpendicular, and it has its slopes equal to the opposite of the inverse of the slope of the other line, so:

slope = - 2

The point (x,y) also pertences to it, such as the point (3,5), so:

-2 = (y - 5)/(x - 3)

y - 5 = -2x +6

y = -2x + 11

So, the turn point is the point where these two lines meet:

x/2 = -2x + 11

x = -4x + 22

5x = 22

x = 22/5

y = (22/5)/2

y = 22/10 = 11/5

So, the point that he makes the turn is (22/5, 11/5).

solution:

→c= a b , that is product of a and b is possible only when when a is m × n matrix and b is n × p matrix.

that is, number of columns of a= number of rows of b.

so, size of matrix , c= a b is , if a = (m × n)matrix

b= (n × p)matrix

then a b = [ (m × n)matrix] ×[   (n × p)matrix ]

→→c= a b= [m × p ] matrix

→→size of product a b will be equal to = number of rows of a × number of columns of b

sas (side angle side)

aa(angle angle)

sss(side side side)

sas you have to find two sides with an angle in between

aa you find two angles because it will always add up to 180

sss find all sides

step-by-step explanation: