# The line l has equation r=(1 2 -1) + lamda(2 1 3). The plane has equation r.(2 -1 -1)=6 (i) show that l is parallel to p Can someone please solve this question?

Jan 28, 2018

See explanation.

#### Explanation:

The direction vector of the line is $\left[2 , 1 , 3\right]$. That vector is parallel to the line.

The normal vector to the plane is $\left[2 , - 1 , - 1\right]$. That vector, as the name implies, is normal to the plane.

Calculating the dot product of the two vectors gives:

$\left[2 , 1 , 3\right] \cdot \left[2 , - 1 , - 1\right] = 4 - 1 - 3 = 0$

Since the dot product is 0 we know the vectors are orthogonal.

Since both the plane and the line are orthogonal to the vector $\left[2 , - 1 , - 1\right]$ they are parallel to each other.