If v= (3, 3, 6) and u = (2, -1, 1), then the length of the projection of u along v is:Select one:a. 3b. 0c. d. e.

The projection of vector u onto vector v is given by the formula:

proj_v(u) = ((u.v)/||v||^2) * v

where:

• u.v is the dot product of u and v,
• ||v|| is the magnitude of v.

First, calculate the dot product of u and v:

u.v = (23) + (-13) + (1*6) = 6 - 3 + 6 = 9

Next, calculate the magnitude of v:

||v|| = sqrt((3^2) + (3^2) + (6^2)) = sqrt(9 + 9 + 36) = sqrt(54)

Then, square the magnitude:

||v||^2 = (sqrt(54))^2 = 54

Now, calculate the projection of u onto v:

proj_v(u) = (u.v/||v||^2) * v = (9/54) * v = (1/6) * v

The length of the projection of u onto v is the magnitude of this projection:

||proj_v(u)|| = ||(1/6) * v|| = (1/6) * ||v|| = (1/6) * sqrt(54) = sqrt(54)/6

So, none of the provided options (a. 3, b. 0, c. , d. , e. ) match the correct answer, sqrt(54)/6.

The projection of vector u onto vector v is given by the formula:

proj_v(u) = ((u.v)/||v||^2) * v

where "." denotes the dot product, and ||v|| denotes the magnitude of vector v.

First, we calculate the dot product of u and v:

u.v = (23) + (-13) + (1*6) = 6 - 3 + 6 = 9

Next, we calculate the magnitude of vector v:

||v|| = sqrt((3^2) + (3^2) + (6^2)) = sqrt(9 + 9 + 36) = sqrt(54)

Then, we square the magnitude:

||v||^2 = (sqrt(54))^2 = 54

Now, we can calculate the projection:

proj_v(u) = (9/54) * v = (1/6) * v = (1/2, 1/2, 1)

The length of the projection of u along v is the magnitude of the projection vector:

||proj_v(u)|| = sqrt((1/2)^2 + (1/2)^2 + 1^2) = sqrt(1/4 + 1/4 + 1) = sqrt(1.5)

So, the length of the projection of u along v is sqrt(1.5), which is not listed in the given options. Therefore, there seems to be a mistake in the question or the provided options.