Let u1(1/√3,1/√3,-1/√3)
Find the projection of A=(a1,a2,a3) on u1
To find the projection of vector A=(a1,a2,a3) onto vector u1=(1/√3,1/√3,-1/√3), we can use the formula for projecting a vector onto another vector:
proj_u1(A) = ((A•u1)/(u1•u1)) * u1
First, let's calculate A•u1:
A•u1 = a1*(1/√3) + a2*(1/√3) + a3*(-1/√3)
= (a1 + a2 - a3)/√3
Next, let's calculate u1•u1:
u1•u1 = (1/√3)^2 + (1/√3)^2 + (-1/√3)^2
= 1/3 + 1/3 + 1/3
= 1
Now we can calculate the projection of A onto u1:
proj_u1(A) = ((A•u1)/(u1•u1)) * u1
= ((a1 + a2 - a3)/√3) * (1/√3,1/√3,-1/√3)
= ((a1 + a2 - a3)/3, (a1 + a2 - a3)/3, -(a1 + a2 - a3)/3)
So, the projection of A=(a1,a2,a3) onto u1=(1/√3,1/√3,-1/√3) is ((a1 + a2 - a3)/3, (a1 + a2 - a3)/3, -(a1 + a2 - a3)/3).
To find the projection of vector A=(a1,a2,a3) on u1=(1/√3,1/√3,-1/√3), we can use the following formula:
Projection of vector A on u1 = (A · u1) * u1
where · represents the dot product between two vectors.
Step 1: Calculate the dot product (A · u1):
(A · u1) = (a1, a2, a3) · (1/√3, 1/√3, -1/√3)
= a1*(1/√3) + a2*(1/√3) + a3*(-1/√3)
Step 2: Multiply the dot product by u1:
Projection of A on u1 = (a1*(1/√3) + a2*(1/√3) + a3*(-1/√3)) * (1/√3, 1/√3, -1/√3)
= ( (a1/√3) + (a2/√3) - (a3/√3) ) * (1/√3, 1/√3, -1/√3)
= ((a1 + a2 - a3) / 3) * (1,1,-1)
Therefore, the projection of vector A=(a1, a2, a3) on u1=(1/√3, 1/√3, -1/√3) is ((a1 + a2 - a3) / 3) * (1,1,-1).