Final answer:
To find the orthogonal projection of the vector [1; 1; 1] onto the line given by scalar multiples of [2; 1; 2], we calculate the dot product of the vectors and apply the formula for orthogonal projection, resulting in [10/9; 5/9; 10/9].
Step-by-step explanation:
The question is asking for the orthogonal projection of the vector [1; 1; 1] onto the line L in ℝ³ that consists of all scalar multiples of [2; 1; 2]. To find this projection, we will use the formula for the orthogonal projection of a vector b onto another vector a, which is (b·a/a·a) * a, where '·' denotes the dot product.
First, we calculate the dot product of [1; 1; 1] and [2; 1; 2], which is 1*2 + 1*1 + 1*2 = 5. Then we calculate the dot product of [2; 1; 2] with itself, which is 2*2+1*1+2*2=9. The projection is then given by (5/9) * [2; 1; 2].
Thus, the orthogonal projection of [1; 1; 1] onto the line L is (5/9)[2; 1; 2] = [10/9; 5/9; 10/9].