The distance between a point P whose position vector is 5 i + j +3 k and the line r =(3 i +7 j + k )+t( j + k ) is

Step-by-step solution

P =5 i + j +3 k P (5,1,3) r =(3 i +7 j + k )+( j + k ) Line passes through the point Q(3,7,1) . P Q =-2 i +6 j -2 k Direction vector of line is ( j + k ) . Distance = | P Q ( j + k )| | j + k | = 1 2 | array ccc i & j & k -2 & 6 & -2 0 & 1 & 1 array | aligned & = | 1 2 [ i (6+2)- j (-2)+ k (-2)] | & = | 1 2 (8 i +2 j -2 k ) | & = 2 |4 i + j - k |= 2 4^2+1^2+1^2 d & = 2 18 =6 aligned