Let L1 L2 and L3 are three lines defined as follows L1 9x y

Let L_1, L_2 and L_3 are three lines defined as follows: L_1: 9x - y = 1, L_2: 3x + y = 5, L_3: 10x - y = 2. Find the distance between the line L_2 and the intersection point of L_1 and L_3. Let a = (3, 4), b = (1, 2) are vectors. Find the unit vector that points in the direction as a. Give a vector a bar of length 3 that points in the direction opposite to a. Calculate pro j_a b.

Solution

Point of intersection of L1 and L2 : 3x +y =5

                                                  10x - y =2

                                                 -------------- 13x = 7 ; x = 7/13

y = 5 -3x = 5 -21/13 = ( 65 -21)/13 = 44/13

( 7/13 , 44/13)

Distance from L : 3x + y =5

[3x1 + y1 -5]/sqrt(9 +1)

([ 3*7/13 + 44/13 -5]/sqrt(10)

b 1) a = ( -3, 4)

Unit vector = (-3, 4)/sqrt(9 +16) = ( -3 , 4)/5 = ( -3/5 , 4/5)

2) unit vector opposite to unit vector a = ( 3/5 , -4/5)

Vector a = 3( 3/5 , -4/5) = ( 9 /5 , -12/5)