Line `L_1` is parallel to vector ` vecalpha=-3 hat i+2 hat j+4 hat k` and passes through a point `A(7,6,2)` and line `L_2` is parallel vector ` vecbeta=2 hat i+ hat j+3 hat k` and point `B(5,3,4)dot` Now a line `L_3` parallel to a vector ` vec r=2 hat i-2 hat j- hat k` intersects the lines `L_1a n dL_2` at points `Ca n dD ,` respectively, then find `| vec C D|dot`

Explanation

Line `L_(1)` is parallel to vector `vecalpha=-3hati+2hatj+4hatk` and passes through a point `A(7, 6, 2)`.
Therefore, position vector of any point on the line is `7hati+6hatj+2hatk+lamda(-3hati+2hatj+4hatk),` line `L_(2)` is parallel to a vector `vecbeta=2hati+hatj+3hatk` and passes through a point `B(5, 3, 4)`.
Position vector of any point on the line is `5hati+3hatj+4hatk+mu(2hati+hatj+3hatk)`
`therefore" "vec(CD)=2hati+3hatj-2hatk+lamda(-3hati+2hatj+4hatk)-mu(2hati=hatj+3hatk)`
Since it is parallel to `2hati-2hatj-hatk`, we have
`" "(2-3lamda-2mu)/(2)=(3+2lamda-mu)/(-2)=(-2+4lamda-3mu)/(-1)`
Solving these equations we get `lamda=2 and mu=1`. Therefore,
`" "vec(CD)=-6hati+6hatj+3hatk`
or `" "|vec(CD)|=9`